#10423134

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

4300 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)(3,9)(4,6)(5,12)(7,11)(8,14)(10,15)(13,16)(17,18,21,24)(19,25,26,22)(20,23,29,28)(27,31,32,30)(33,34,38,41)(35,39,36,43)(37,40,46,45)(42,47,44,48);(2,4)(7,9)(8,12)(15,16)(19,26)(22,25)(27,32)(30,31)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,16)(13,14)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32)(33,35,38,36)(34,39,41,43)(37,42,46,44)(40,47,45,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,34)(35,39)(36,43)(37,45)(38,41)(40,46)(42,48)(44,47);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32)(33,36,38,35)(34,39,41,43)(37,44,46,42)(40,47,45,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,31)(30,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10330330);