#303057

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
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

705 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)(11,15)(12,20)(13,21)(14,16)(17,18)(19,24)(22,23)(25,26,30,34)(27,31,37,40)(28,32,29,33)(35,38,36,39);(1,3,5,4)(2,6,8,7)(9,11,14,19)(10,15,16,24)(12,17,21,23)(13,22,20,18)(25,27)(26,31)(28,36)(29,35)(30,37)(32,39)(33,38)(34,40);(9,12,20,10,14,21,13,16)(11,17,22,24,19,23,18,15)(27,35)(28,29)(31,38)(32,33)(36,37)(39,40);(1,4,5,3)(2,7,8,6)(25,28,30,29)(26,32,34,33)(27,36,37,35)(31,39,40,38)

Polycyclic presentation:

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

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, 210253);