#8630913

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	#53518	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(33,34,38,41)(35,37,40,47)(36,43,46,39)(42,48,44,45);(2,7)(3,11)(4,14)(13,15)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(33,35,38,40)(34,37,41,47)(36,44,46,42)(39,48,43,45);(2,7)(4,14)(8,10)(12,16)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,36,38,46)(34,39,41,43)(35,42,40,44)(37,45,47,48);(1,3,6,11)(2,8,7,10)(4,12,14,16)(5,15,9,13)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,35)(34,37)(36,42)(38,40)(39,45)(41,47)(43,48)(44,46);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,43)(47,48);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,37)(34,40)(35,41)(36,45)(38,47)(39,44)(42,43)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=g,e^2=ghi,f^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, 8538109);