#8905243

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

4227 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8812439);