#364710

order	512
nilpotency class	3
isoclinism	isoclinic class of #196344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1662 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22,25,16)(13,17)(14,18)(19,30,23,27)(20,28)(21,29)(24,35,37,31)(26,32)(33,40,36,38)(34,39);(2,5)(6,8)(9,11)(10,15)(12,23,25,19)(13,20)(14,21)(16,30,22,27)(17,28)(18,29)(24,36,37,33)(26,34)(31,40,35,38)(32,39);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,24)(14,25)(15,27)(17,31)(18,22)(20,33)(21,23)(26,37)(28,38)(29,30)(32,35)(34,36)(39,40);(2,5)(6,8)(9,13,14,26)(10,17,18,32)(11,20,21,34)(12,24,25,37)(15,28,29,39)(16,31,22,35)(19,33,23,36)(27,38,30,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=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, 271906);