#224518

order	512
nilpotency class	3
isoclinism	isoclinic class of #196038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(2,6)(7,8)(9,10)(11,17)(12,20,21,15)(13,16)(14,19)(18,22,24,23)(25,26,30,34,29,33,28,32)(27,35,36,40,37,31,38,39);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,29,37)(26,31,33,35)(28,38,30,36)(32,40,34,39);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,28,29,30)(26,32,33,34)(27,36,37,38)(31,39,35,40);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(22,23)(27,37)(31,35)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=eh,d^2=ehi,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,g^c=gh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 131714);