#8436675

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	#54602	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 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)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(18,23)(19,26)(25,30)(27,32)(33,34)(35,40)(36,47)(37,41)(38,42)(39,43)(44,45)(46,48);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(17,18,22,23)(19,21,26,24)(20,25,29,30)(27,28,32,31)(33,35,39,46)(34,40,43,48)(36,37,42,44)(38,45,47,41);(17,19)(18,21)(20,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,36,39,42)(34,38,43,47)(35,44,46,37)(40,41,48,45);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,30)(29,32)(33,35)(34,40)(36,44)(37,42)(38,41)(39,46)(43,48)(45,47);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,20,22,29)(18,25,23,30)(19,27,26,32)(21,28,24,31)(33,37,39,44)(34,41,43,45)(35,42,46,36)(38,48,47,40);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,21,22,24)(18,26,23,19)(20,28,29,31)(25,32,30,27)(33,38)(34,42)(35,45)(36,43)(37,48)(39,47)(40,44)(41,46)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,e^2=h,f^2=g,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, 8343871);