#7537886

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1948 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di

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, 7445082);