#8533409

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
prime radical	#56411	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4141 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=i,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8440605);