#6871567

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1647 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)(3,6)(4,5)(7,8)(10,13)(11,15)(17,18,22,26,21,25,35,42)(19,27,30,44,29,23,37,39)(20,31,34,41,33,43,36,24)(28,45,46,38,32,48,47,40);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,32)(21,29)(22,30)(24,40)(25,27)(26,39)(28,33)(31,45)(34,47)(35,37)(36,46)(38,41)(42,44)(43,48);(10,13)(14,16)(17,20,21,33)(18,24,25,41)(19,28,29,32)(22,36,35,34)(23,38,27,40)(26,43,42,31)(30,47,37,46)(39,45,44,48);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(18,25)(19,29)(24,41)(26,42)(28,32)(30,37)(31,43)(46,47);(1,3,4,7)(2,6,5,8)(18,25)(23,27)(24,41)(26,42)(31,43)(38,40)(39,44)(45,48)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=hi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6778763);