#392341

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1672 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)(7,12)(8,10)(9,11)(13,14)(15,17)(16,18)(19,20)(21,22)(23,26)(24,27)(25,29)(28,32)(30,33)(31,34)(35,36);(5,7,9,8)(6,10,11,12)(13,15,16,19)(14,17,18,20)(21,23)(22,26)(24,31)(25,30)(27,34)(28,35)(29,33)(32,36);(5,8,9,7)(6,10,11,12)(14,18)(17,20)(23,30)(26,33)(31,36)(34,35);(22,27)(24,32)(26,34)(28,29)(31,36)(33,35)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,c^b=ci,g^d=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299537);