#8127228

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

4176 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)(21,29)(23,30)(26,32)(28,31)(33,34)(35,44)(36,46)(37,40)(38,48)(39,43)(41,47)(42,45);(1,3,6,9)(2,7,8,14)(4,10,13,15)(5,11,12,16)(17,18,22,24)(19,25,27,20)(21,23,29,30)(26,31,32,28)(33,35,39,37)(34,40,43,44)(36,45,47,38)(41,42,46,48);(2,8)(3,9)(4,13)(11,16)(17,19,22,27)(18,20,24,25)(21,26,29,32)(23,28,30,31)(33,34)(35,40)(36,46)(37,44)(38,42)(39,43)(41,47)(45,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,36,39,47)(34,41,43,46)(35,45,37,38)(40,42,44,48);(1,4)(2,5)(3,10)(6,13)(7,11)(8,12)(9,15)(14,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,37,39,35)(34,40,43,44)(36,38,47,45)(41,42,46,48);(1,5)(2,4)(3,11)(6,12)(7,10)(8,13)(9,16)(14,15)(17,21)(18,23)(19,26)(20,28)(22,29)(24,30)(25,31)(27,32)(33,38)(34,42)(35,36)(37,47)(39,45)(40,46)(41,44)(43,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=g,e^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, 8034424);