#10340247

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	#41052	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4046 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10247443);