#8691733

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

4224 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=gh,e^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, 8598929);