#6792667

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 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)(19,28,30,23)(20,32,34,25)(22,24,36,27)(29,38,33,41)(31,39,44,37)(35,42,46,40)(43,47,45,48);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,13)(14,16)(17,19,21,30)(18,23,26,28)(20,33,34,29)(22,31,36,44)(24,37,27,39)(25,41,32,38)(35,45,46,43)(40,48,42,47);(3,8)(5,7)(10,13)(14,15)(18,24)(22,36)(23,37)(25,40)(26,27)(28,39)(31,44)(32,42)(35,46)(38,47)(41,48)(43,45);(2,6)(5,7)(17,20,21,34)(18,25,26,32)(19,29,30,33)(22,35,36,46)(23,38,28,41)(24,40,27,42)(31,43,44,45)(37,47,39,48);(9,11,12,16)(10,14,13,15)(18,26)(19,30)(20,34)(24,27)(31,44)(35,46)(38,41)(47,48)

Polycyclic presentation:

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

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, 6699863);