#9002444

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	#44236	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 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)(3,11)(4,15)(5,10)(6,7)(8,14)(9,13)(12,16)(17,18,22,25)(19,23,26,21)(20,24,28,31)(27,29,32,30)(33,34)(35,40)(36,39)(37,41)(38,42)(43,45)(44,47)(46,48);(1,3,7,14)(2,8,6,11)(4,12,13,5)(9,16,15,10)(17,19)(18,23)(20,27)(21,25)(22,26)(24,29)(28,32)(30,31)(33,35)(34,36)(37,43)(38,39)(40,42)(41,44)(45,48)(46,47);(2,6)(4,13)(5,12)(8,11)(19,26)(20,28)(21,23)(24,31)(33,36,38,40)(34,39,42,35)(37,45,46,44)(41,43,48,47);(1,4)(2,9)(3,12)(5,14)(6,15)(7,13)(8,16)(10,11)(17,20)(18,24)(19,27)(21,30)(22,28)(23,29)(25,31)(26,32)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,46)(47,48);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,15,11,9)(17,21,22,23)(18,19,25,26)(20,29,28,30)(24,32,31,27)(33,35)(34,40)(36,42)(37,43)(38,39)(41,45)(44,48)(46,47);(1,6,7,2)(3,11,14,8)(4,15,13,9)(5,16,12,10)(17,19)(18,23)(20,27)(21,25)(22,26)(24,29)(28,32)(30,31)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,45)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8909640);