#360224

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,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, 267420);