#370006

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1308 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=eg,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277202);