#302805

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
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

643 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)(11,18)(12,22)(13,16)(14,17)(15,21)(19,20)(23,24)(25,26)(27,32)(28,29)(30,37)(31,33)(34,35)(36,40)(38,39);(1,3,5,4)(2,6,8,7)(9,10)(11,19)(12,16)(13,22)(14,17)(15,21)(18,20)(23,24)(25,27)(26,32)(28,34)(29,35)(30,38)(31,36)(33,40)(37,39);(9,11,13,23,14,20,12,21)(10,15,22,19,17,24,16,18)(25,28,29,26,31,37,30,33)(27,34,38,40,36,39,35,32);(1,4,5,3)(2,7,8,6)(11,20)(15,24)(18,19)(21,23)(25,29,31,30)(26,28,33,37)(27,35,36,38)(32,34,40,39)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,d^2=ei,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,d^c=dh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 210001);