#9007215

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	#55084	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=gh,e^2=i,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, 8914411);