#7564560

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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

1979 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)(13,14,18,23)(15,24,26,20)(16,19,22,17)(21,28,27,25);(1,2)(3,4)(6,10)(11,12)(13,15,18,26)(14,20,23,24)(16,27,22,21)(17,25,19,28);(5,7,9,8)(6,11,10,12)(13,16,19,23,18,22,17,14)(15,24,25,27,26,20,28,21);(2,4)(13,15,19,25,18,26,17,28)(14,21,16,24,23,27,22,20);(5,8,9,7)(6,11,10,12)(13,17,18,19)(14,22,23,16)(15,25,26,28)(20,21,24,27)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=g,e^2=f,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,e^d=ei

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, 7471756);