#6547150

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1646 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:

(2,4)(21,22,26,30,27,31,25,39)(23,28,35,43,33,42,34,49)(24,36,37,45,38,44,40,29)(32,46,48,51,47,52,50,41);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(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);(6,10)(11,17)(12,18)(19,20)(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);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(23,33)(28,42)(32,47)(34,35)(41,51)(43,49)(46,52)(48,50);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(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=g,b^2=fi,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh

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, 6454346);