#10041472

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
prime radical	#43586	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 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)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(19,26)(24,27)(28,31)(29,32)(33,34,38,41)(35,37,43,39)(36,40,46,48)(42,47,44,45);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(17,18,22,23)(19,27,26,24)(20,21,25,30)(28,32,31,29)(33,35,38,43)(34,39,41,37)(36,44,46,42)(40,47,48,45);(2,8)(4,9)(7,14)(10,15)(18,23)(20,25)(24,27)(28,31)(33,35,38,43)(34,39,41,37)(36,44,46,42)(40,47,48,45);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,19,22,26)(18,24,23,27)(20,28,25,31)(21,29,30,32)(33,35)(34,37)(36,42)(38,43)(39,41)(40,47)(44,46)(45,48);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,20)(18,21)(19,28)(22,25)(23,30)(24,29)(26,31)(27,32)(33,36)(34,40)(35,42)(37,47)(38,46)(39,45)(41,48)(43,44);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,21)(18,25)(19,29)(20,23)(22,30)(24,31)(26,32)(27,28)(33,37)(34,35)(36,45)(38,39)(40,44)(41,43)(42,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9948668);