#472452

order	512
nilpotency class	3
isoclinism	isoclinic class of #453960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379648);