#8695882

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	#55094	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 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)(4,9)(5,10)(6,7)(8,12)(13,14)(15,16)(18,19)(20,27)(24,26)(28,29)(33,34,39,37)(35,40,46,44)(36,47,45,41)(38,42,43,48);(2,7)(3,12)(9,14)(10,16)(17,18,22,19)(20,23,27,25)(21,24,30,26)(28,31,29,32)(33,35,39,46)(34,40,37,44)(36,38,45,43)(41,48,47,42);(17,18,22,19)(20,23,27,25)(21,24,30,26)(28,31,29,32)(33,36)(34,41)(35,43)(37,47)(38,46)(39,45)(40,42)(44,48);(1,3)(2,8)(4,10)(5,14)(6,12)(7,11)(9,15)(13,16)(17,19,22,18)(20,25,27,23)(21,24,30,26)(28,31,29,32)(33,37,39,34)(35,44,46,40)(36,47,45,41)(38,42,43,48);(1,4)(2,9)(3,10)(5,11)(6,13)(7,14)(8,15)(12,16)(17,20)(18,23)(19,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,36)(34,41)(35,38)(37,47)(39,45)(40,48)(42,44)(43,46);(1,5,6,15)(2,10,7,16)(3,9,12,14)(4,8,13,11)(17,21,22,30)(18,24,19,26)(20,28,27,29)(23,31,25,32)(33,38)(34,42)(35,45)(36,46)(37,48)(39,43)(40,41)(44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603078);