#374302

order	512
nilpotency class	3
isoclinism	isoclinic class of #196266
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281498);