#379639

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄³

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

1134 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)(21,22)(23,30)(24,28)(25,27)(26,31)(29,33)(32,35)(34,36);(6,9)(10,12)(13,14)(15,18)(16,17)(19,20)(21,23,24,31)(22,26,28,30)(25,32,33,36)(27,34,29,35);(5,7,8,11)(6,10,9,12)(14,17)(18,20)(22,27)(25,33)(26,34)(28,29)(30,35)(32,36);(1,3,4,2)(6,9)(10,12)(13,15,16,19)(14,18,17,20)(22,28)(23,31)(27,29)(32,36)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ei,d^2=gh,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, 286835);