#10444892

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

4140 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)(3,7)(4,12)(5,9)(6,11)(8,13)(10,14)(15,16)(17,18,23,20)(19,24,21,26)(22,25,30,28)(27,31,29,32)(33,34)(35,37)(36,40)(38,41)(39,42)(43,45)(44,46)(47,48);(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,32)(29,30)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,36,39,37)(34,40,42,35)(38,44,47,45)(41,46,48,43);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,20,23,18)(19,26,21,24)(22,25,30,28)(27,31,29,32)(33,35)(34,36)(37,42)(38,46)(39,40)(41,45)(43,47)(44,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,37,39,36)(34,40,42,35)(38,45,47,44)(41,46,48,43);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,46)(42,48)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=i,e^2=gi,f^2=h,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, 10352088);