#6676871

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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

1999 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:

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

Polycyclic presentation:

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

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, 6584067);