#7503251

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

2661 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)(3,7)(4,6)(5,8)(9,10,14,20,15,21,16,22)(11,23,26,40,27,17,35,36)(12,28,31,37,30,25,29,18)(13,24,34,19,33,39,32,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,19)(13,28)(14,26)(15,27)(16,35)(18,32)(20,36)(21,23)(22,40)(24,29)(25,33)(30,38)(31,39)(34,37);(9,12)(10,18)(11,24)(13,23)(14,29)(15,30)(16,31)(17,33)(19,35)(20,25)(21,37)(22,28)(26,38)(27,39)(32,40)(34,36);(2,6)(3,8)(9,13)(10,19)(11,25)(12,23)(14,32)(15,33)(16,34)(17,30)(18,35)(20,24)(21,38)(22,39)(26,37)(27,28)(29,40)(31,36);(9,14,15,16)(10,20,21,22)(11,26,27,35)(12,29,30,31)(13,32,33,34)(17,36,23,40)(18,25,37,28)(19,24,38,39)

Polycyclic presentation:

2^9|a^2=g,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, 7410447);