#405914

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1550 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:

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

Polycyclic presentation:

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

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, 313110);