#7616323

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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
prime radical	C₂×#127	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1454 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)(3,7)(4,5)(6,8)(11,12)(14,15)(17,18)(19,26)(20,28)(21,25)(22,23)(24,30)(27,31)(29,32);(2,5)(6,7)(9,10,13,16)(11,15,12,14)(19,22)(20,29)(23,26)(24,31)(27,30)(28,32);(9,11,13,12)(10,14,16,15)(17,19,21,22)(18,23,25,26)(20,29,30,27)(24,32,28,31);(9,12,13,11)(10,15,16,14)(17,20,22,29,21,30,19,27)(18,24,23,31,25,28,26,32);(1,3,4,8)(2,6,5,7)(20,30)(24,28)(27,29)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340627, #72340663, #72340697
maximal normal subgroup of	#72340627, #72340663, #72340697

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7523519);