#10422714

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

4031 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10329910);