#302759

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

648 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)(9,10,13,18)(11,17,20,16)(12,22,21,24)(14,23,15,19)(25,26,31,35)(27,32,33,38)(28,37,39,36)(29,34,30,40);(1,3,5,4)(2,6,8,7)(9,11,13,20)(10,16,18,17)(12,14,21,15)(19,24,23,22)(25,27)(26,32)(28,34)(29,36)(30,37)(31,33)(35,38)(39,40);(9,12,14,11,13,21,15,20)(10,17,19,22,18,16,23,24)(25,28,29,38,31,39,30,32)(26,33,40,36,35,27,34,37);(1,4,5,3)(2,7,8,6)(10,18)(16,17)(19,23)(22,24)(25,29,31,30)(26,34,35,40)(27,36,33,37)(28,38,39,32)

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=fi,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209955);