#8505350

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 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)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(34,39)(36,44)(40,42)(43,45);(1,3)(2,8)(4,11)(5,6)(7,13)(9,15)(10,14)(12,16)(18,20)(21,26)(25,28)(29,32)(33,34,38,39)(35,40,37,42)(36,41,44,46)(43,47,45,48);(1,2,7,4)(3,11,13,8)(5,9,16,14)(6,10,12,15)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,32)(30,31)(33,35,38,37)(34,40,39,42)(36,43,44,45)(41,47,46,48);(1,4,7,2)(3,11,13,8)(5,14,16,9)(6,10,12,15)(17,20,23,18)(19,26,24,21)(22,25,30,28)(27,29,31,32)(33,36)(34,41)(35,43)(37,45)(38,44)(39,46)(40,47)(42,48);(1,5)(2,9)(3,12)(4,14)(6,13)(7,16)(8,10)(11,15)(17,21,23,26)(18,24,20,19)(22,29,30,32)(25,31,28,27)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,6,7,12)(2,10,4,15)(3,5,13,16)(8,14,11,9)(17,22,23,30)(18,25,20,28)(19,27,24,31)(21,29,26,32)(33,37,38,35)(34,42,39,40)(36,43,44,45)(41,47,46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=hi,e^2=h,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, 8412546);