#7170530

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	#42788	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1587 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,c^2=i,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7077726);