#8253876

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(17,18,23,22)(19,26,25,21)(20,24,28,30)(27,32,31,29)(34,35)(36,43)(40,42)(44,48);(1,2)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(20,28)(24,30)(27,31)(29,32)(33,34,38,35)(36,41,43,39)(37,40,45,42)(44,47,48,46);(2,6)(7,9)(8,12)(14,15)(17,19)(18,21)(20,27)(22,26)(23,25)(24,29)(28,31)(30,32)(33,35,38,34)(36,41,43,39)(37,40,45,42)(44,46,48,47);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20,23,28)(18,24,22,30)(19,27,25,31)(21,29,26,32)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21,23,26)(18,25,22,19)(20,29,28,32)(24,31,30,27)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,22,23,18)(19,26,25,21)(20,30,28,24)(27,32,31,29)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8161072);