#8720593

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=i,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627789);