#381290

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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 ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,3,4)(6,9)(10,12)(14,17)(18,19)(21,22)(23,29)(24,26)(25,33)(27,31)(28,32)(30,36)(34,35);(5,6,8,9)(7,10,11,12)(13,14,16,17)(15,19,20,18)(22,26)(27,35)(28,33)(29,36);(7,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,23,24,30)(22,27,26,35)(25,34,32,31)(28,36,33,29);(5,7,8,11)(6,10,9,12)(15,20)(18,19)(23,30)(27,35)(29,36)(31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288486);