#9054512

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

4115 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)(17,18,23,20)(19,26,25,22)(21,24,29,28)(27,32,31,30)(33,34,39,43)(35,40,44,38)(36,37,42,41)(45,47,48,46);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31)(33,35,39,44)(34,40,43,38)(36,45,42,48)(37,47,41,46);(2,5)(3,4)(6,14)(15,16)(18,20)(21,29)(22,26)(27,31)(35,44)(36,42)(37,41)(38,40);(1,4,7,3)(2,8,5,10)(6,12,14,11)(9,15,13,16)(17,20,23,18)(19,26,25,22)(21,24,29,28)(27,30,31,32)(33,36)(34,41)(35,45)(37,43)(38,47)(39,42)(40,46)(44,48);(1,5,7,2)(3,10,4,8)(6,13,14,9)(11,16,12,15)(17,21,23,29)(18,24,20,28)(19,27,25,31)(22,30,26,32)(33,37,39,41)(34,42,43,36)(35,46,44,47)(38,48,40,45);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,22,23,26)(18,25,20,19)(21,30,29,32)(24,31,28,27)(33,38)(34,35)(36,47)(37,48)(39,40)(41,45)(42,46)(43,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=h,e^2=gi,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, 8961708);