#387174

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1048 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=egi,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,f^c=fi,f^d=fi

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, 294370);