#9016999

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	#55312	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4229 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)(3,9)(4,12)(5,6)(7,11)(8,14)(10,15)(13,16)(17,18,22,25)(19,26,20,23)(21,24,30,29)(27,32,28,31)(33,34,38,43)(35,41,44,39)(36,40,42,37)(45,46,47,48);(2,6)(7,9)(8,12)(15,16)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(35,44)(39,41)(45,47)(46,48);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,18,22,25)(19,23,20,26)(21,29,30,24)(27,32,28,31)(33,35,38,44)(34,39,43,41)(36,47,42,45)(37,48,40,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,16)(13,14)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,36,38,42)(34,40,43,37)(35,45,44,47)(39,48,41,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,35)(34,41)(36,47)(37,46)(38,44)(39,43)(40,48)(42,45);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)(33,37,38,40)(34,42,43,36)(35,46,44,48)(39,47,41,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=i,e^2=ghi,f^2=gh,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, 8924195);