#8587450

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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)(3,5,12,8)(4,13,16,10)(6,11,14,15)(17,18,23,26)(19,24,27,21)(20,22,25,31)(28,29,32,30)(33,34,38,41)(35,39,43,37)(36,44,47,40)(42,46,45,48);(1,3,7,12)(2,8,9,5)(4,14,16,6)(10,15,13,11)(17,19)(18,24)(20,28)(21,26)(22,29)(23,27)(25,32)(30,31)(33,35)(34,39)(36,45)(37,41)(38,43)(40,46)(42,47)(44,48);(2,9)(5,8)(10,13)(11,15)(19,27)(21,24)(28,32)(29,30)(33,36,38,47)(34,40,41,44)(35,42,43,45)(37,46,39,48);(1,4,7,16)(2,10,9,13)(3,6,12,14)(5,15,8,11)(17,20)(18,22)(19,28)(21,30)(23,25)(24,29)(26,31)(27,32)(33,37)(34,35)(36,46)(38,39)(40,42)(41,43)(44,45)(47,48);(1,5,7,8)(2,3,9,12)(4,15,16,11)(6,13,14,10)(17,21)(18,19)(20,30)(22,28)(23,24)(25,29)(26,27)(31,32)(33,36)(34,40)(35,42)(37,46)(38,47)(39,48)(41,44)(43,45);(1,6)(2,11)(3,4)(5,13)(7,14)(8,10)(9,15)(12,16)(17,22)(18,25)(19,29)(20,26)(21,28)(23,31)(24,32)(27,30)(33,35)(34,39)(36,42)(37,41)(38,43)(40,48)(44,46)(45,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=g,e^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, 8494646);