#8755415

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	#46690	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(19,26)(22,24)(27,32)(29,31)(33,34)(35,37)(36,44)(38,39)(40,42)(41,46)(43,47)(45,48);(2,7)(5,8)(9,12)(13,15)(17,18,23,20)(19,24,26,22)(21,28,30,25)(27,29,32,31)(34,39)(35,42)(41,44)(43,48);(2,7)(3,10)(9,12)(11,16)(17,19,23,26)(18,24,20,22)(21,27,30,32)(25,31,28,29)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,20,23,18)(19,22,26,24)(21,28,30,25)(27,29,32,31)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,21,23,30)(18,25,20,28)(19,27,26,32)(22,29,24,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,22)(18,19)(20,26)(21,29)(23,24)(25,27)(28,32)(30,31)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662611);