#10412331

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

4272 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,e^2=g,f^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, 10319527);