#8605702

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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)(3,4)(5,13)(6,10)(7,8)(9,16)(11,14)(12,15)(17,18)(19,25)(20,26)(21,24)(22,23)(27,31)(28,32)(29,30)(34,38)(35,41)(40,46)(43,44);(1,3,7,11)(2,4,8,14)(5,15,16,6)(9,10,13,12)(18,23)(19,26)(24,30)(27,32)(33,34,37,38)(35,42,41,39)(36,40,45,46)(43,48,44,47);(2,8)(3,11)(5,16)(10,12)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,34,37,38)(35,39,41,42)(36,40,45,46)(43,47,44,48);(1,4)(2,3)(5,12)(6,9)(7,14)(8,11)(10,16)(13,15)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,35)(34,39)(36,44)(37,41)(38,42)(40,48)(43,45)(46,47);(1,5)(2,9)(3,6)(4,12)(7,16)(8,13)(10,14)(11,15)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,48)(45,46);(1,6)(2,10)(3,5)(4,13)(7,15)(8,12)(9,14)(11,16)(17,21,22,29)(18,24,23,30)(19,27,26,32)(20,28,25,31)(33,36,37,45)(34,40,38,46)(35,43,41,44)(39,47,42,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8512898);