#10464553

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

4318 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10371749);