#6772407

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,d^2=fhi,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,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6679603);