#304844

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
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

702 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)(9,10,15,11)(12,20,23,24)(13,17,14,19)(16,21,18,22)(25,26)(27,33)(28,39)(29,34)(30,36)(31,35)(32,37)(38,40);(1,3,5,4)(2,6,8,7)(9,11,15,10)(12,18,23,16)(13,17,14,19)(20,22,24,21)(25,27)(26,32)(28,30)(29,37)(31,38)(33,34)(35,39)(36,40);(1,3,5,4)(2,7,8,6)(9,12,13,21,15,23,14,22)(10,16,19,24,11,18,17,20)(25,28,30,27,29,38,31,37)(26,33,35,39,34,32,36,40);(1,4,5,3)(2,7,8,6)(9,13,15,14)(10,17,11,19)(12,21,23,22)(16,20,18,24)(26,34)(27,37)(28,38)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212040);