#6966447

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1420 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(17,18,21,26,22,25,35,40)(19,27,29,37,30,41,34,23)(20,31,33,47,32,24,38,39)(28,43,45,36,44,42,48,46);(1,3,4,8)(2,5,6,7)(10,13)(11,15);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,25,37)(20,28,32,44)(21,34,35,29)(24,36,31,46)(26,41,40,27)(33,48,38,45)(39,43,47,42);(2,6)(5,7)(10,13)(14,16)(17,20,22,32)(18,24,25,31)(19,28,30,44)(21,33,35,38)(23,36,37,46)(26,39,40,47)(27,42,41,43)(29,45,34,48);(9,11,12,15)(10,14,13,16)(18,25)(20,32)(23,37)(26,40)(27,41)(28,44)(33,38)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6873643);