#7195092

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	#44877	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fghi,d^2=i,e^2=hi,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=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, 7102288);