#7600933

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fg,c^2=ghi,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251233, #72256827, #72256920, …
maximal normal subgroup of	#72251233, #72256827, #72256920, …
maximal quotient of	#72308872

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7508129);