#7108258

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#42778	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1344 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)(18,22)(19,23)(24,27)(25,31)(26,30)(28,32);(1,2)(3,8)(4,7)(5,6)(10,11)(12,13)(17,18,21,22)(19,23,26,30)(20,27,29,24)(25,32,28,31);(9,11,14,10)(12,15,13,16)(17,19,21,26)(18,23,22,30)(20,28,29,25)(24,32,27,31);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,15,11,16)(17,20)(18,24)(19,25)(21,29)(22,27)(23,31)(26,28)(30,32);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,15,11,16)(20,29)(24,27)(25,28)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=fi,e^2=fi,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=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7015454);