#323304

order	512
nilpotency class	3
isoclinism	isoclinic class of #298603
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)(9,10,14,22,12,15,13,17)(11,18,21,16,19,24,20,23)(25,26,29,33,28,32,38,31)(27,34,37,40,36,30,35,39);(1,3,5,4)(2,6,8,7)(10,15)(16,23)(17,22)(18,24)(25,27,29,37,28,36,38,35)(26,30,33,39,32,34,31,40);(9,11,12,19)(10,16,15,23)(13,21,14,20)(17,24,22,18)(26,31)(27,35)(29,38)(30,34)(32,33)(36,37);(1,4,5,3)(2,7,8,6)(10,15)(16,23)(17,22)(18,24)(26,32)(30,34)(31,33)(39,40)

Polycyclic presentation:

2^9|a^2=efi,b^2=egh,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230500);