#420323

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2535	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1888 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=ei,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327519);