#10444915

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

4156 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=i,d^2=gi,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, 10352111);