#8639766

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	#45708	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=g,e^2=g,f^2=gi,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, 8546962);