#297189

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
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

602 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=f,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=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204385);