#382121

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

1112 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ef,d^2=efhi,g^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=de,g^c=gi

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, 289317);