#8505559

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	#57307	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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)(3,11)(4,13)(5,10)(6,7)(8,12)(9,14)(15,16)(17,18,23,20)(19,24,29,27)(21,30,32,25)(22,26,28,31)(35,43)(36,45)(39,46)(40,48);(2,7)(4,14)(8,11)(10,16)(17,19)(18,24)(20,27)(21,22)(23,29)(25,31)(26,30)(28,32)(33,34,38,42)(35,39,43,46)(36,40,45,48)(37,41,44,47);(17,18,23,20)(19,27,29,24)(21,25,32,30)(22,26,28,31)(33,35)(34,39)(36,44)(37,45)(38,43)(40,47)(41,48)(42,46);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,13,15,9)(17,20,23,18)(19,27,29,24)(21,30,32,25)(22,26,28,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4)(2,9)(3,10)(5,8)(6,14)(7,13)(11,15)(12,16)(17,21)(18,25)(19,28)(20,30)(22,29)(23,32)(24,26)(27,31)(33,36,38,45)(34,40,42,48)(35,44,43,37)(39,47,46,41);(1,5)(2,10)(3,9)(4,8)(6,15)(7,16)(11,14)(12,13)(17,22,23,28)(18,26,20,31)(19,21,29,32)(24,30,27,25)(33,37,38,44)(34,41,42,47)(35,45,43,36)(39,48,46,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8412755);