#7594820

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#45417	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1622 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,25)(19,26,29,23)(20,21,32,27)(24,28,31,30);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,31)(21,28)(22,29)(24,32)(25,26)(27,30);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,21,25,22,32,27,18)(19,26,30,31,29,23,28,24);(9,11,12,16)(10,14,13,15)(17,19,27,28,22,29,21,30)(18,24,32,23,25,31,20,26);(2,5)(6,8)(17,21,22,27)(18,20,25,32)(19,28,29,30)(23,31,26,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7502016);