#8633703

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8540899);