#7562131

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2306 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)(7,11)(8,12)(9,10)(13,14)(15,18)(16,19)(17,24)(20,22)(21,27)(23,26)(25,28);(1,2)(3,4)(6,10)(11,12)(13,15,20,16)(14,18,22,19)(17,25,26,21)(23,27,24,28);(5,7,9,8)(6,11,10,12)(13,16,19,14,20,15,18,22)(17,24,28,21,26,23,27,25);(2,4)(5,8,9,7)(6,12,10,11)(13,17,18,27,20,26,19,28)(14,21,16,24,22,25,15,23);(6,10)(11,12)(17,26)(21,25)(23,24)(27,28)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=fg,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=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei

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, 7469327);