#6676226

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
prime radical	#30454	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1868 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)(5,6,9,11)(7,12,8,10)(13,14,18,19)(15,20,24,28)(16,25,23,21)(17,22,27,26);(5,6)(7,10)(8,12)(9,11)(13,15,18,24)(14,20,19,28)(16,22,23,26)(17,25,27,21);(2,4)(13,16,18,23)(14,21,19,25)(15,17,24,27)(20,26,28,22);(5,7,9,8)(6,10,11,12)(13,14,18,19)(15,20,24,28)(16,21,23,25)(17,26,27,22);(5,8,9,7)(6,10,11,12)(13,17)(14,22)(15,23)(16,24)(18,27)(19,26)(20,21)(25,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=fi,e^2=f,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, 6583422);