#8496811

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8404007);