#8127258

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

4257 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)(4,12)(5,13)(6,8)(9,14)(10,16)(11,15)(18,23)(19,25)(21,29)(28,31)(33,34)(35,44)(36,37)(38,41)(39,40)(42,43)(45,48)(46,47);(1,3,6,9)(2,7,8,14)(4,10,13,15)(5,11,12,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(34,40)(36,43)(41,44)(45,47);(2,8)(3,9)(4,13)(11,16)(19,25)(20,24)(27,32)(28,31)(33,35,39,38)(34,41,40,44)(36,47,43,45)(37,48,42,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19,22,25)(18,24,23,20)(21,27,29,32)(26,31,30,28)(33,36,39,43)(34,42,40,37)(35,45,38,47)(41,48,44,46);(1,4)(2,5)(3,10)(6,13)(7,11)(8,12)(9,15)(14,16)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,37,39,42)(34,43,40,36)(35,46,38,48)(41,47,44,45);(1,5)(2,4)(3,11)(6,12)(7,10)(8,13)(9,16)(14,15)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,38,39,35)(34,44,40,41)(36,47,43,45)(37,48,42,46)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=g,e^2=g,f^2=g,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, 8034454);