#7041792

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#44975	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 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)(9,10,13,16)(11,14,12,15)(17,18)(19,28)(20,22)(21,26)(23,27)(24,32)(25,31)(29,30);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,25,27,28)(20,29,32,26)(21,24,30,22);(9,11)(10,14)(12,13)(15,16)(17,20,22,18,23,32,24,27)(19,29,21,25,31,26,30,28);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,21,23,30)(18,26,27,29)(19,22,31,24)(20,25,32,28);(10,16)(14,15)(17,22,23,24)(18,20,27,32)(19,30,31,21)(25,26,28,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6948988);