#472128

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

982 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379324);