#7592359

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
prime radical	#44915	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1672 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)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,23,26)(19,22,25,21)(20,27,32,29)(24,30,28,31);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,28)(21,26)(23,25)(24,32)(27,31)(29,30);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,21,30,23,32,22,31)(18,24,25,29,26,28,19,27);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,19,22,26,23,25,21,18)(20,29,31,24,32,27,30,28);(2,6)(7,8)(10,14)(13,15)(17,21,23,22)(18,19,26,25)(20,30,32,31)(24,27,28,29)

Polycyclic presentation:

2^9|a^2=ghi,c^2=fgi,d^2=fg,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,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7499555);