#6541154

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(2,3)(5,6)(7,16)(8,13)(9,14)(10,12)(11,15)(17,20)(18,19)(21,22,27,31)(23,28,33,42)(24,36,38,44)(25,30,26,39)(29,40,45,37)(32,46,47,52)(34,43,35,49)(41,50,51,48);(1,2,4,3)(5,7,11,10)(6,12,15,16)(8,18,9,17)(13,20,14,19)(21,23,27,33)(22,28,31,42)(24,32,38,47)(25,34,26,35)(29,41,45,51)(30,43,39,49)(36,46,44,52)(37,48,40,50);(1,3,4,2)(5,8,11,9)(6,13,15,14)(7,17,10,18)(12,19,16,20)(21,24,27,38)(22,29,31,45)(23,32,33,47)(25,37,26,40)(28,41,42,51)(30,44,39,36)(34,48,35,50)(43,52,49,46);(5,9,11,8)(6,14,15,13)(7,17,10,18)(12,19,16,20)(23,33)(28,42)(32,47)(34,35)(41,51)(43,49)(46,52)(48,50);(5,10,11,7)(6,12,15,16)(8,18,9,17)(13,19,14,20)(21,25,27,26)(22,30,31,39)(23,34,33,35)(24,37,38,40)(28,43,42,49)(29,44,45,36)(32,48,47,50)(41,52,51,46)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6448350);