#9054509

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4139 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961705);