#8633782

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	#49786	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4101 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=h,f^2=hi,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, 8540978);