#323623

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=egi,c^2=i,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230819);