#8506282

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

4256 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)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(20,28)(24,29)(27,32)(30,31)(33,34,39,43)(35,44,46,40)(36,37,45,41)(38,42,48,47);(3,10)(4,12)(6,7)(8,15)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,30)(31,32)(33,35,39,46)(34,40,43,44)(36,38,45,48)(37,47,41,42);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31)(33,34,39,43)(35,40,46,44)(36,41,45,37)(38,42,48,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,36)(34,41)(35,38)(37,43)(39,45)(40,42)(44,47)(46,48);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,20,22,28)(18,24,26,29)(19,27,25,32)(21,30,23,31)(33,37,39,41)(34,36,43,45)(35,42,46,47)(38,40,48,44);(1,2)(3,6)(4,8)(5,9)(7,10)(11,13)(12,15)(14,16)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,32,29,27)(33,38)(34,42)(35,45)(36,46)(37,40)(39,48)(41,44)(43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=ghi,f^2=hi,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, 8413478);