#8630830

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	#56559	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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)(3,8,13,15)(4,10,11,5)(6,12,16,14)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(33,34,38,37)(35,41,43,39)(36,44,46,40)(42,47,45,48);(1,3)(2,8)(4,14)(5,16)(6,10)(7,13)(9,15)(11,12)(18,23)(19,26)(25,28)(27,32)(33,35,38,43)(34,39,37,41)(36,45,46,42)(40,48,44,47);(2,9)(3,13)(5,10)(12,14)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,35,38,43)(34,39,37,41)(36,42,46,45)(40,47,44,48);(1,4,7,11)(2,10,9,5)(3,12,13,14)(6,8,16,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,36)(34,40)(35,42)(37,44)(38,46)(39,47)(41,48)(43,45);(1,5,7,10)(2,11,9,4)(3,6,13,16)(8,14,15,12)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,48)(45,46);(1,6)(2,12)(3,10)(4,15)(5,13)(7,16)(8,11)(9,14)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,37,38,34)(35,41,43,39)(36,40,46,44)(42,47,45,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=hi,d^2=gh,e^2=i,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, 8538026);