#6676643

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#42876	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1960 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)(8,12)(9,11)(13,14,18,22)(15,17,21,27)(16,26,19,20)(23,24,28,25);(5,7,10,6)(8,12,9,11)(13,14)(15,23)(16,20)(17,25)(18,22)(19,26)(21,28)(24,27);(2,4)(13,15,18,21)(14,17,22,27)(16,25,19,24)(20,28,26,23);(5,8,10,9)(6,11,7,12)(13,16,18,19)(14,20,22,26)(15,24,21,25)(17,23,27,28);(5,9,10,8)(6,11,7,12)(13,17)(14,21)(15,22)(16,23)(18,27)(19,28)(20,25)(24,26)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6583839);