#6763219

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#39356	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1772 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=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, 6670415);