#6524265

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1463 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6431461);