#360225

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
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

1540 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)(5,6,9,15,10,16,25,14)(7,12,18,11,20,27,19,28)(8,21,23,13,24,31,36,32)(17,33,34,29,22,26,35,30);(5,7,9,18,10,20,25,19)(6,11,14,12,16,28,15,27)(8,17,23,34,24,22,36,35)(13,26,21,29,32,33,31,30);(2,4)(5,7)(6,12)(8,22)(9,18)(10,20)(11,15)(13,30)(14,28)(16,27)(17,24)(19,25)(21,26)(23,35)(29,32)(31,33)(34,36);(5,8,9,23,10,24,25,36)(6,13,14,21,16,32,15,31)(7,17,18,34,20,22,19,35)(11,26,12,29,28,33,27,30)

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,d^c=di,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, 267421);