#421549

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1092 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:

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

Polycyclic presentation:

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

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, 328745);