#294786

order	512
nilpotency class	3
isoclinism	isoclinic class of #271003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

825 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)(3,7)(4,5)(6,8)(9,10,14,17)(11,15,13,22)(12,19,21,24)(16,18,23,20)(25,26,30,35)(27,33,37,32)(28,39,38,40)(29,36,31,34);(2,5)(6,7)(9,11,14,13)(10,15,17,22)(12,18,21,20)(16,19,23,24)(25,27,30,37)(26,32,35,33)(28,31,38,29)(34,39,36,40);(9,12,11,18,14,21,13,20)(10,16,22,24,17,23,15,19)(25,28,31,37,30,38,29,27)(26,33,36,40,35,32,34,39);(1,3,4,8)(2,6,5,7)(25,29,30,31)(26,34,35,36)(27,28,37,38)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 201982);