#8188037

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	#43191	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(18,19)(20,27)(24,26)(28,32)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(2,7)(3,10)(9,14)(11,16)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,35,38,46)(34,39,42,43)(36,37,45,44)(40,41,47,48);(5,13)(9,14)(11,16)(12,15)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,48,43,41);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,14)(13,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(1,4,6,8)(2,3,7,10)(5,12,13,15)(9,11,14,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095233);