#8095482

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	#52636	C₂
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002678);