#7501210

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2295 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,14,17,21)(15,19,24,27)(16,25,18,20)(22,23,28,26);(5,7)(6,10)(8,11)(9,12)(13,15,17,24)(14,19,21,27)(16,23,18,26)(20,22,25,28);(2,4)(5,8,9,10)(6,7,11,12)(13,14)(15,22)(16,20)(17,21)(18,25)(19,26)(23,27)(24,28);(2,4)(6,11)(7,12)(13,14)(15,19)(16,25)(17,21)(18,20)(22,26)(23,28)(24,27);(13,16,17,18)(14,20,21,25)(15,23,24,26)(19,22,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408406);