#8097246

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	#53076	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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)(3,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,34,39,38)(35,40,45,44)(36,46,37,41)(42,48,43,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,39,45)(34,40,38,44)(36,42,37,43)(41,47,46,48);(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,36,39,37)(34,41,38,46)(35,42,45,43)(40,47,44,48);(1,4,5,13)(2,7,9,12)(3,10,11,16)(6,14,8,15)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,35)(34,40)(36,42)(37,43)(38,44)(39,45)(41,47)(46,48);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,21,23,19)(18,24,20,26)(22,27,30,29)(25,32,28,31)(33,37,39,36)(34,41,38,46)(35,43,45,42)(40,47,44,48);(1,2)(3,6)(4,7)(5,9)(8,11)(10,14)(12,13)(15,16)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,38,39,34)(35,44,45,40)(36,46,37,41)(42,48,43,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004442);