#6484618

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473174
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6391814);