#8810329

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	#45801	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3996 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)(3,6)(4,12)(5,8)(7,14)(9,11)(10,16)(13,15)(18,20)(19,27)(22,32)(25,29)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,16,12,15)(17,18,23,20)(19,28,27,24)(21,29,31,25)(22,26,32,30)(36,44)(39,46)(42,48)(45,47);(1,2,5,8)(3,9,11,6)(4,7,14,12)(10,16,13,15)(17,19)(18,24)(20,28)(21,22)(23,27)(25,26)(29,30)(31,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,20,23,18)(19,28,27,24)(21,29,31,25)(22,30,32,26);(1,3)(2,6)(4,10)(5,11)(7,15)(8,9)(12,16)(13,14)(17,21)(18,25)(19,22)(20,29)(23,31)(24,26)(27,32)(28,30)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48);(1,4,5,14)(2,7,8,12)(3,10,11,13)(6,15,9,16)(17,22)(18,26)(19,21)(20,30)(23,32)(24,25)(27,31)(28,29)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=g,f^2=gi,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, 8717525);