#9946994

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	#41249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4105 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,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, 9854190);