#8139501

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,12)(4,5)(6,14)(7,11)(8,16)(9,15)(10,13)(17,18,21,25)(19,23,26,22)(20,28,29,24)(27,30,32,31)(33,34,39,44)(35,43,38,40)(36,42,37,41)(45,47,46,48);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(19,26)(22,23)(27,32)(30,31)(33,35,39,38)(34,40,44,43)(36,45,37,46)(41,47,42,48);(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,36,39,37)(34,41,44,42)(35,45,38,46)(40,47,43,48);(1,4,7,15)(2,9,11,5)(3,13,16,6)(8,10,12,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,37,39,36)(34,42,44,41)(35,45,38,46)(40,47,43,48);(1,5,7,9)(2,4,11,15)(3,14,16,10)(6,8,13,12)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,38,39,35)(34,43,44,40)(36,46,37,45)(41,48,42,47);(1,6,7,13)(2,10,11,14)(3,15,16,4)(5,8,9,12)(17,20,21,29)(18,24,25,28)(19,27,26,32)(22,30,23,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,47)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8046697);