#10422683

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

4027 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)(3,4)(5,9)(6,10)(7,8)(11,12)(13,15)(14,16)(18,19)(20,27)(21,29)(30,31)(33,34,38,41)(35,39,44,37)(36,40,46,48)(42,47,43,45);(1,2,6,10)(3,7,8,4)(5,9,14,16)(11,15,13,12)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30)(33,35,38,44)(34,39,41,37)(36,43,46,42)(40,45,48,47);(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,3)(2,7)(4,10)(5,13)(6,8)(9,12)(11,14)(15,16)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,36)(34,40)(35,43)(37,47)(38,46)(39,45)(41,48)(42,44);(1,4,6,7)(2,8,10,3)(5,12,14,15)(9,13,16,11)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,35)(34,37)(36,42)(38,44)(39,41)(40,45)(43,46)(47,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,37)(34,35)(36,45)(38,39)(40,42)(41,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=h,e^2=gh,f^2=h,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, 10329879);