#8621810

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=h,e^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, 8529006);