#6535572

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,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, 6442768);