#384009

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1070 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=defi,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, 291205);