#7005756

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	#42926	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1610 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,28,23,27,22,26)(19,20,29,30,21,25,31,32);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,27)(21,23)(22,31)(24,29)(26,30)(28,32);(9,11)(10,13)(12,16)(14,15)(17,20,23,25)(18,19,27,21)(22,30,24,32)(26,29,28,31);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21,23,19)(18,25,27,20)(22,29,24,31)(26,30,28,32);(10,14)(13,15)(17,22,23,24)(18,26,27,28)(19,29,21,31)(20,30,25,32)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,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, 6912952);