#8551142

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	#54053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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:

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

Polycyclic presentation:

2^9|a^2=gi,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, 8458338);