#8522135

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	#54442	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18,23,26)(19,24,27,21)(20,22,25,31)(28,29,32,30)(33,34,39,35)(36,43,46,40)(37,41,38,42)(44,48,45,47);(1,3,6,11)(2,7,4,9)(5,10,14,13)(8,15,12,16)(17,19,23,27)(18,24,26,21)(20,28,25,32)(22,29,31,30)(34,35)(40,43)(41,42)(47,48);(2,4)(7,9)(8,12)(15,16)(19,27)(21,24)(28,32)(29,30)(33,35,39,34)(36,43,46,40)(37,42,38,41)(44,48,45,47);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,20)(18,22)(19,28)(21,30)(23,25)(24,29)(26,31)(27,32)(33,36,39,46)(34,40,35,43)(37,44,38,45)(41,47,42,48);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,21)(18,19)(20,30)(22,28)(23,24)(25,29)(26,27)(31,32)(33,37,39,38)(34,41,35,42)(36,44,46,45)(40,47,43,48);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,22)(18,25)(19,29)(20,26)(21,28)(23,31)(24,32)(27,30)(33,38,39,37)(34,42,35,41)(36,45,46,44)(40,48,43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=gh,e^2=g,f^2=g,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, 8429331);