#7623668

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623668
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1898 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)(5,25)(6,14)(7,13)(8,11)(9,19)(10,23)(12,29)(16,48)(17,53)(18,47)(20,57)(22,59)(24,43)(26,44)(27,41)(28,39)(30,42)(31,37)(32,40)(33,54)(34,49)(35,61)(36,56)(38,60)(45,51)(46,55)(50,58)(52,63)(62,64);(1,3)(2,9)(4,16)(5,26)(6,20)(7,32)(8,18)(10,33)(11,36)(12,40)(13,35)(14,44)(15,25)(17,22)(19,29)(21,58)(23,51)(24,49)(27,50)(28,56)(30,61)(31,52)(34,38)(37,53)(39,57)(41,45)(42,47)(43,46)(48,59)(54,60)(55,62)(63,64);(1,4)(2,10)(3,16)(5,22)(6,31)(7,23)(8,24)(9,33)(11,37)(12,38)(13,43)(14,21)(15,45)(17,26)(18,49)(19,50)(20,52)(25,41)(27,29)(28,62)(30,60)(32,51)(34,40)(35,46)(36,53)(39,59)(42,64)(44,58)(47,63)(48,57)(54,61)(55,56);(1,4)(2,10)(3,17)(5,27)(6,23)(7,31)(8,24)(9,34)(11,37)(12,41)(13,21)(14,43)(15,46)(16,26)(18,54)(19,52)(20,50)(22,29)(25,38)(28,60)(30,62)(32,55)(33,40)(35,45)(36,48)(39,64)(42,59)(44,63)(47,58)(49,61)(51,56)(53,57);(2,11)(3,18)(10,37)(12,39)(13,14)(16,49)(17,54)(19,56)(20,32)(21,43)(25,42)(26,61)(38,59)(41,64)(50,55)(51,52)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285648
maximal normal subgroup of	#72285648
maximal quotient of	#72342602, #72342604

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530864);