#10247485

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	#57157	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4183 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34,38,40)(35,36,43,39)(37,45,47,41)(42,48,46,44);(2,3)(7,9)(8,10)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,43)(34,39,40,36)(37,42,47,46)(41,48,45,44);(18,22)(23,25)(24,28)(30,31)(34,40)(36,39)(41,45)(44,48);(1,3,6,2)(4,7,12,9)(5,10,14,8)(11,15,13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,38,39)(34,35,40,43)(37,44,47,48)(41,42,45,46);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,35)(34,36)(37,46)(38,43)(39,40)(41,48)(42,47)(44,45);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,37)(34,41)(35,42)(36,44)(38,47)(39,48)(40,45)(43,46)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154681);