#8695847

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

4191 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603043);