#8504947

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3882 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8412143);