#10534344

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	#49862	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3936 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10441540);