#323610

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

620 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)(9,10,15,23,13,16,14,17)(11,18,24,21,19,12,20,22)(25,26,30,32,28,33,29,34)(27,35,36,39,37,31,38,40);(1,3,5,4)(2,6,8,7)(9,11)(10,12)(13,19)(14,20)(15,24)(16,18)(17,21)(22,23)(25,27,30,36,28,37,29,38)(26,31,32,40,33,35,34,39);(9,12,13,18)(10,11,16,19)(14,22,15,21)(17,24,23,20)(26,32)(27,36)(29,30)(31,35)(33,34)(37,38);(1,4,5,3)(2,7,8,6)(10,16)(11,19)(17,23)(20,24)(26,33)(27,37)(32,34)(36,38)

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,c^2=hi,d^2=eh,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, 230806);