#7108071

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#45257	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7015267);