#7503458

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1760 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)(3,4)(5,7)(6,8)(9,10,15,21,16,22,14,20)(11,17,26,35,27,36,25,34)(12,24,28,37,29,40,33,18)(13,23,30,39,32,38,31,19);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,19)(13,24)(14,25)(15,26)(16,27)(18,31)(20,34)(21,35)(22,36)(23,28)(29,39)(30,37)(32,40)(33,38);(1,4,5,6)(2,3,7,8)(9,12)(10,18)(11,23)(13,17)(14,28)(15,33)(16,29)(19,26)(20,24)(21,40)(22,37)(25,39)(27,38)(30,34)(31,35)(32,36);(2,7)(3,8)(9,13)(10,19)(11,24)(12,17)(14,30)(15,31)(16,32)(18,26)(20,23)(21,38)(22,39)(25,37)(27,40)(28,34)(29,36)(33,35);(9,14,16,15)(10,20,22,21)(11,25,27,26)(12,28,29,33)(13,30,32,31)(17,34,36,35)(18,24,37,40)(19,23,39,38)

Polycyclic presentation:

2^9|a^2=g,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi

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, 7410654);