#6493917

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37422	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1539 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)(11,15)(14,16)(17,18,23,28)(19,29,33,43)(20,34,35,48)(21,26,37,41)(22,39,30,27)(24,40,42,32)(25,31,44,36)(38,46,47,45);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,30,40,23,33,22,32)(18,24,39,43,28,42,27,29)(20,21,31,47,35,37,36,38)(25,41,48,45,44,26,34,46);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,22,36,23,35,30,31)(18,25,27,34,28,44,39,48)(19,21,32,38,33,37,40,47)(24,41,29,46,42,26,43,45);(10,13)(14,16)(17,21)(18,26)(19,31)(20,32)(22,38)(23,37)(24,34)(25,43)(27,45)(28,41)(29,44)(30,47)(33,36)(35,40)(39,46)(42,48);(9,11,12,15)(10,14,13,16)(20,35)(21,37)(25,44)(26,41)(31,36)(34,48)(38,47)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,e^2=h,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,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, 6401113);