#475980

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

948 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383176);