#7564871

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2017 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=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, 7472067);