#7041618

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1404 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)(19,26)(21,22)(24,31)(25,30)(27,32)(28,29);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,20)(19,27,29,24)(21,30,22,25)(26,32,28,31);(9,11)(10,14)(12,13)(15,16)(17,19,21,28,23,29,22,26)(18,24,30,32,20,27,25,31);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,20,23,18)(19,24,29,27)(21,25,22,30)(26,31,28,32);(10,16)(14,15)(17,21,23,22)(18,25,20,30)(19,28,29,26)(24,31,27,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gi,d^2=fi,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=eh,d^c=dh,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6948814);