#8505107

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

4253 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)(3,7)(4,8)(5,9)(6,10)(11,15)(12,13)(14,16)(17,18,23,20)(19,24,29,27)(21,30,32,25)(22,26,28,31)(33,34,39,37)(35,42,40,36)(38,41,47,45)(43,46,48,44);(1,3,6,9)(2,7,10,5)(4,12,14,11)(8,13,16,15)(17,19)(18,24)(20,27)(21,22)(23,29)(25,31)(26,30)(28,32)(34,37)(36,42)(41,45)(44,46);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,14)(15,16)(17,18,23,20)(19,27,29,24)(21,25,32,30)(22,26,28,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,18)(19,27,29,24)(21,30,32,25)(22,26,28,31)(33,36,39,42)(34,40,37,35)(38,46,47,44)(41,43,45,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21)(18,25)(19,28)(20,30)(22,29)(23,32)(24,26)(27,31)(33,37,39,34)(35,42,40,36)(38,45,47,41)(43,46,48,44);(1,5,6,7)(2,9,10,3)(4,13,14,15)(8,12,16,11)(17,22,23,28)(18,26,20,31)(19,21,29,32)(24,30,27,25)(33,38,39,47)(34,41,37,45)(35,43,40,48)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=hi,e^2=g,f^2=gi,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, 8412303);