#6816478

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#42477	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1851 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:

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

Polycyclic presentation:

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

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, 6723674);