#6484709

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1660 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,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, 6391905);