#6587771

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1535 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,21,26)(19,25,30,23)(20,32,35,24)(22,27,37,45)(28,39,47,43)(29,40,31,44)(33,41,34,48)(36,46,38,42);(2,6)(7,8)(17,19)(18,23)(20,33)(21,30)(22,29)(24,41)(25,26)(27,44)(28,36)(31,37)(32,48)(34,35)(38,47)(39,42)(40,45)(43,46);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,21,35)(18,24,26,32)(19,28,30,47)(22,38,37,36)(23,39,25,43)(27,46,45,42)(29,33,31,34)(40,48,44,41);(9,11,12,15)(10,14,13,16)(17,19)(18,25)(20,34)(21,30)(22,29)(23,26)(24,41)(27,40)(28,38)(31,37)(32,48)(33,35)(36,47)(39,42)(43,46)(44,45);(1,4,5,3)(2,8,6,7)(18,26)(19,29)(22,37)(23,40)(24,32)(25,44)(28,33)(30,31)(34,47)(36,38)(39,48)(41,43)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6494967);