#8471040

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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)(3,11)(4,9)(5,6)(7,10)(8,15)(12,14)(13,16)(18,23)(19,26)(25,30)(27,32)(33,34,39,36)(35,40,46,43)(37,47,45,41)(38,42,44,48);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,18,22,23)(19,21,26,24)(20,25,29,30)(27,28,32,31)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,42)(47,48);(1,4,7,5)(2,9,10,6)(3,12,15,13)(8,16,11,14)(17,19)(18,21)(20,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,34,39,36)(35,43,46,40)(37,41,45,47)(38,42,44,48);(1,5,7,4)(2,9,10,6)(3,13,15,12)(8,16,11,14)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,30)(29,32)(33,36,39,34)(35,43,46,40)(37,47,45,41)(38,42,44,48);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,20,22,29)(18,25,23,30)(19,27,26,32)(21,28,24,31)(33,37,39,45)(34,41,36,47)(35,44,46,38)(40,42,43,48);(1,6,7,9)(2,5,10,4)(3,14,15,16)(8,13,11,12)(17,21,22,24)(18,26,23,19)(20,28,29,31)(25,32,30,27)(33,38,39,44)(34,42,36,48)(35,45,46,37)(40,41,43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=ghi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8378236);