#8127620

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

4222 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(33,34,37,41)(35,39,42,38)(36,44,45,40)(43,46,48,47);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(35,42)(38,39)(43,48)(46,47);(2,5)(4,14)(8,10)(11,13)(19,26)(20,28)(22,25)(23,30)(33,35,37,42)(34,38,41,39)(36,43,45,48)(40,46,44,47);(1,4)(2,9)(3,11)(5,12)(6,13)(7,14)(8,15)(10,16)(17,19)(18,22)(20,27)(21,26)(23,29)(24,25)(28,32)(30,31)(33,35)(34,39)(36,43)(37,42)(38,41)(40,47)(44,46)(45,48);(1,5,7,2)(3,10,6,8)(4,12,14,9)(11,16,13,15)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,6,7,3)(2,10,5,8)(4,13,14,11)(9,16,12,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,36,37,45)(34,40,41,44)(35,43,42,48)(38,46,39,47)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8034816);