#6668797

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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 ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2226 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,h^e=hi

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, 6575993);