#10439965

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	#40842	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3981 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)(4,12)(5,9)(6,11)(8,13)(10,14)(15,16)(17,18,23,28)(19,26,21,24)(20,25,27,22)(29,30,31,32)(34,39)(36,43)(41,45)(44,48);(19,21)(24,26)(29,31)(30,32)(33,34)(35,36)(37,41)(38,39)(40,43)(42,44)(45,46)(47,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,19,23,21)(18,24,28,26)(20,29,27,31)(22,30,25,32)(33,35,38,40)(34,36,39,43)(37,42,46,47)(41,44,45,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,20)(18,25)(19,29)(21,31)(22,28)(23,27)(24,32)(26,30)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,48)(44,47);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,21,23,19)(18,26,28,24)(20,31,27,29)(22,32,25,30)(33,36)(34,40)(35,39)(37,44)(38,43)(41,47)(42,45)(46,48);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,22,23,25)(18,27,28,20)(19,30,21,32)(24,31,26,29)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=i,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, 10347161);