#7622952

order	512
nilpotency class	4
isoclinism	isoclinic class of #33595
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#459
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂	C₂²×#459
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2023 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)(3,15)(4,21)(5,11)(6,26)(7,12)(8,14)(9,18)(10,23)(13,28)(16,43)(17,33)(19,34)(20,32)(22,53)(24,54)(25,40)(27,35)(29,38)(30,39)(31,49)(36,57)(37,55)(41,46)(42,50)(44,62)(45,52)(47,60)(48,56)(51,61)(58,64)(59,63);(1,3,5,18,8,17,7,20)(2,9,12,15,14,32,11,33)(4,16,22,45,24,44,30,51)(6,19,25,48,29,47,28,50)(10,31,36,41,37,58,35,59)(13,34,38,42,40,60,26,56)(21,52,39,43,54,61,53,62)(23,46,27,49,55,63,57,64);(1,4)(2,10)(3,16)(5,22)(6,27)(7,30)(8,24)(9,31)(11,35)(12,36)(13,39)(14,37)(15,41)(17,44)(18,45)(19,49)(20,51)(21,26)(23,28)(25,55)(29,57)(32,58)(33,59)(34,43)(38,54)(40,53)(42,61)(46,50)(47,64)(48,63)(52,56)(60,62);(3,17)(9,32)(15,33)(16,44)(18,20)(19,47)(31,58)(34,60)(41,59)(42,56)(43,62)(45,51)(46,63)(48,50)(49,64)(52,61);(1,5,8,7)(2,11,14,12)(3,18,17,20)(4,22,24,30)(6,25,29,28)(9,33,32,15)(10,35,37,36)(13,26,40,38)(16,45,44,51)(19,48,47,50)(21,53,54,39)(23,27,55,57)(31,59,58,41)(34,56,60,42)(43,52,62,61)(46,49,63,64)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249762, #72249769, #72249836, …
maximal normal subgroup of	#72249762, #72249769, #72249836, …
maximal quotient of	#72250042, C₂×#7622952, #72323026, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530148);