#10340217

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	#53280	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3949 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=g,e^2=h,f^2=g,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, 10247413);