#497651

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, prime radical, cosocle	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#554 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=ei,d^2=e,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fi,g^c=ghi,g^f=gi

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, 404847);