#7582262

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1677 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,22)(21,30)(23,25)(26,29)(27,28)(31,32);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,26)(21,28)(22,29)(23,31)(25,32)(27,30);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,18,22,20)(19,26,29,24)(21,30,23,25)(27,28,32,31);(10,14)(13,15)(17,20,22,18)(19,27,29,32)(21,30,23,25)(24,31,26,28);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,25,20,30)(19,28,29,31)(24,32,26,27)

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257381, #72257477, #72264023
maximal normal subgroup of	#72257381, #72257477, #72264023
maximal quotient of	#72308378

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7489458);