#10166043

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

4054 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18,24,27)(19,28,31,25)(20,32,22,23)(21,29,30,26)(33,34,40,44)(35,39,37,42)(36,46,45,41)(38,43,47,48);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19,24,31)(18,25,27,28)(20,29,22,26)(21,23,30,32)(33,35,40,37)(34,39,44,42)(36,47,45,38)(41,43,46,48);(2,4)(3,10)(9,12)(11,16)(17,20,24,22)(18,23,27,32)(19,29,31,26)(21,25,30,28)(33,36,40,45)(34,41,44,46)(35,38,37,47)(39,48,42,43);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,21,24,30)(18,26,27,29)(19,23,31,32)(20,28,22,25)(33,37,40,35)(34,42,44,39)(36,38,45,47)(41,48,46,43);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,22,24,20)(18,23,27,32)(19,29,31,26)(21,28,30,25)(33,38)(34,43)(35,45)(36,37)(39,41)(40,47)(42,46)(44,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,23)(18,22)(19,30)(20,27)(21,31)(24,32)(25,26)(28,29)(33,39,40,42)(34,35,44,37)(36,48,45,43)(38,46,47,41)

Polycyclic presentation:

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

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, 10073239);