#9778466

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	#55028	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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)(5,9)(6,13)(7,8)(10,15)(11,16)(34,37)(39,41)(40,45)(46,47);(2,8)(3,5)(10,14)(11,13)(17,18,23,26)(19,20,21,24)(22,25,30,32)(27,28,29,31)(36,44)(40,45)(42,48)(46,47);(1,3,7,5)(2,9,8,12)(4,11,15,13)(6,10,16,14)(17,19,23,21)(18,24,26,20)(22,27,30,29)(25,31,32,28)(33,34,38,37)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,4,7,15)(2,10,8,14)(3,13,5,11)(6,12,16,9)(17,20,23,24)(18,19,26,21)(22,28,30,31)(25,27,32,29)(33,35,38,43)(34,39,37,41)(36,42,44,48)(40,46,45,47);(1,5,7,3)(2,9,8,12)(4,11,15,13)(6,14,16,10)(17,21,23,19)(18,24,26,20)(22,27,30,29)(25,28,32,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,6)(2,11)(3,10)(4,12)(5,14)(7,16)(8,13)(9,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,37,38,34)(35,41,43,39)(36,45,44,40)(42,47,48,46)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=ghi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dghi,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9685662);