#10195642

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

4045 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,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, 10102838);