#6498788

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2002 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)(5,6)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22,26,30)(23,31,36,48)(24,28,37,44)(25,39,32,29)(27,40,43,35)(33,46,34,47)(38,52,49,45)(41,51,42,50);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,32,40,26,36,25,35)(22,27,39,48,30,43,29,31)(24,33,49,51,37,34,38,50)(28,41,52,47,44,42,45,46);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(21,24)(22,28)(23,33)(25,38)(26,37)(27,41)(29,45)(30,44)(31,46)(32,49)(34,36)(35,50)(39,52)(40,51)(42,43)(47,48);(7,13)(10,17)(14,19)(18,20)(21,24)(22,28)(23,34)(25,38)(26,37)(27,42)(29,45)(30,44)(31,47)(32,49)(33,36)(35,51)(39,52)(40,50)(41,43)(46,48);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(24,37)(28,44)(33,34)(38,49)(41,42)(45,52)(46,47)(50,51)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6405984);