#6372484

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂×#3455	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=g,e^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303121, #72303551, #72303567
maximal normal subgroup of	#72303121, #72303551, #72303567

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279680);