#8634191

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=g,f^2=gi,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, 8541387);