#6380625

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1956 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)(4,8)(5,7)(10,15,18,25)(16,27,29,35)(17,22,30,36)(28,31,38,39);(1,3,5,4)(2,6,7,8)(9,10)(11,16)(12,22)(13,25)(14,18)(15,26)(17,23)(19,31)(20,29)(21,35)(24,36)(27,34)(28,33)(30,37)(32,39)(38,40);(1,4,5,3)(2,6,7,8)(9,11)(10,16)(12,19)(13,21)(14,20)(15,27)(17,28)(18,29)(22,31)(23,33)(24,32)(25,35)(26,34)(30,38)(36,39)(37,40);(9,12,13,23,14,24,26,37)(10,17,15,22,18,30,25,36)(11,19,21,33,20,32,34,40)(16,28,27,31,29,38,35,39);(11,20)(16,29)(19,32)(21,34)(27,35)(28,38)(31,39)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287821);