#387932

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

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

1532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 295128);