#6527049

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1972 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)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22)(23,29)(24,36)(25,30)(26,41)(27,32)(28,34)(31,37)(33,43)(35,44)(38,49)(39,45)(40,51)(42,48)(46,50)(47,52);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,34,27,35,37,24)(22,28,31,29,32,36,41,44)(25,33,40,48,39,47,50,38)(30,42,46,43,45,49,51,52);(7,13)(8,15)(10,16)(11,18)(21,24,37,35,27,34,26,23)(22,29,41,28,32,44,31,36)(25,38,50,47,39,48,40,33)(30,43,51,42,45,52,46,49);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(21,25)(22,30)(23,33)(24,38)(26,40)(27,39)(28,42)(29,43)(31,46)(32,45)(34,48)(35,47)(36,49)(37,50)(41,51)(44,52);(7,13)(10,16)(14,19)(17,20)(25,39)(30,45)(33,47)(38,48)(40,50)(42,49)(43,52)(46,51)

Polycyclic presentation:

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

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, 6434245);