#7622837

order	512
nilpotency class	4
isoclinism	isoclinic class of #33531
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#447
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂	C₂²×#447
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3845 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,14)(4,10)(5,11)(6,24)(7,29)(8,13)(9,17)(12,27)(15,44)(16,42)(18,43)(19,35)(20,36)(21,51)(22,54)(23,40)(25,32)(26,56)(28,57)(30,38)(31,47)(33,49)(34,59)(37,53)(39,50)(41,45)(46,64)(48,60)(52,61)(55,63)(58,62);(1,3)(2,9)(4,15)(5,16)(6,25)(7,30)(8,18)(10,31)(11,32)(12,38)(13,33)(14,24)(17,27)(19,45)(20,46)(21,52)(22,55)(23,49)(26,58)(28,59)(29,34)(35,60)(36,61)(37,63)(39,64)(40,42)(41,56)(43,57)(44,51)(47,53)(48,50)(54,62);(1,4)(2,10)(3,15)(5,20)(6,26)(7,22)(8,19)(9,31)(11,36)(12,39)(13,35)(14,41)(16,46)(17,48)(18,45)(21,28)(23,53)(24,56)(25,58)(27,50)(29,54)(30,55)(32,61)(33,60)(34,62)(37,40)(38,64)(42,63)(43,44)(47,49)(51,57)(52,59);(4,19)(10,35)(15,45)(20,22)(21,26)(31,60)(36,54)(37,39)(41,44)(46,55)(47,48)(50,53)(51,56)(52,58)(61,62)(63,64);(1,5,8,7)(2,11,13,29)(3,16,18,30)(4,20,19,22)(6,23,28,27)(9,32,33,34)(10,36,35,54)(12,24,40,57)(14,42,43,38)(15,46,45,55)(17,25,49,59)(21,50,26,53)(31,61,60,62)(37,51,39,56)(41,63,44,64)(47,52,48,58)

Polycyclic presentation:

2^9|e^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fh,h^a=hi,f^b=fh,g^b=gi,h^b=hi,d^c=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249757, #72249762, #72249764, …
maximal normal subgroup of	#72249757, #72249762, #72249764, …
maximal quotient of	#72249968, #72250120, C₂×#7622837

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530033);