#8139507

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	#49824	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8046703);