#6677074

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1980 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)(7,10)(8,12)(9,11)(13,14)(15,18)(16,24)(17,22)(19,21)(20,25)(23,27)(26,28);(1,2)(3,4)(5,7,10,6)(8,12,9,11)(13,15,18,14)(16,20,26,23)(17,22,19,21)(24,27,28,25);(2,4)(13,16,18,26)(14,20,15,23)(17,25,19,27)(21,24,22,28);(5,8,10,9)(6,11,7,12)(14,15)(20,23)(21,22)(24,28);(2,4)(5,9,10,8)(6,11,7,12)(13,17,18,19)(14,21,15,22)(16,25,26,27)(20,24,23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6584270);