#363579

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1382 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)(8,10)(11,12)(21,22)(23,30)(24,27)(25,29)(26,32)(28,34)(31,36)(33,35);(5,7,8,11)(6,9,10,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,33)(25,31)(27,30)(28,36)(29,35)(32,34);(6,10)(9,12)(14,17)(18,20)(23,31)(26,35)(30,36)(32,33);(13,15,16,19)(14,18,17,20)(22,27)(24,34)(26,30)(28,29)(32,33)(35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=gi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,e^d=ei

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, 270775);