#321121

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

624 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)(11,14)(12,15)(13,16)(17,24)(18,22)(19,21)(20,23)(25,26,31,35,30,33,29,27)(28,36,39,32,38,34,37,40);(2,5)(6,7)(9,11,13,19,12,18,20,17)(10,14,16,21,15,22,23,24)(25,27,29,33,30,35,31,26)(28,32,37,36,38,40,39,34);(10,15)(11,17)(13,20)(14,21)(18,19)(22,24)(25,28,30,38)(26,32,33,40)(27,34,35,36)(29,39,31,37);(1,3,4,8)(2,6,5,7)(10,15)(14,22)(16,23)(21,24)(26,33)(27,35)(32,40)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228317);