#10419998

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=gh,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, 10327194);