#471932

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

1194 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)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22,25,28,26,27,34,35)(23,24,30,31,29,32,36,33);(1,2,3,4)(6,9)(10,12)(13,15,16,19)(14,17,18,20)(21,23)(22,24)(25,30)(26,29)(27,32)(28,31)(33,35)(34,36);(2,4)(7,11)(10,12)(21,24,26,32)(22,23,27,29)(25,33,34,31)(28,36,35,30);(5,7,8,11)(6,10,9,12)(14,18)(17,20)(22,27)(23,29)(28,35)(30,36)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=i,d^2=e,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, 379128);