#8505546

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

4116 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412742);