#6767537

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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1759 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)(19,30)(20,27)(21,37)(22,28)(23,29)(24,39)(25,33)(26,36)(31,41)(32,46)(34,42)(35,45)(38,44)(40,48)(43,47);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,35)(21,31)(22,32)(23,33)(24,34)(26,43)(27,41)(28,42)(29,30)(36,48)(37,45)(38,47)(39,46)(40,44);(10,13)(14,16)(17,20,23,21)(18,26,29,44)(19,31,33,35)(22,36,24,38)(25,40,30,43)(27,28,37,39)(32,47,34,48)(41,46,45,42);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,23,20)(18,27,29,37)(19,31,33,35)(22,36,24,38)(25,41,30,45)(26,28,44,39)(32,48,34,47)(40,46,43,42);(3,8)(5,7)(10,13)(14,16)(17,22,23,24)(18,28,29,39)(19,32,33,34)(20,36,21,38)(25,42,30,46)(26,37,44,27)(31,47,35,48)(40,41,43,45)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6674733);