#7503820

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2356 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411016);