#8772534

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3855 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8679730);