#287729

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=fi,c^2=h,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,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, 194925);