#7172584

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=f,c^2=fi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7079780);