#10202127

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

4093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=ghi,e^2=gh,f^2=ghi,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, 10109323);