#8551206

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	#54931	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458402);