#9910055

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

4184 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)(3,5)(4,8)(6,9)(7,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,23,27,30)(20,21,28,29)(25,26,31,32)(33,34,39,41)(35,45,46,40)(36,37,43,42)(38,48,47,44);(1,3,7,4)(2,8,10,5)(6,11,15,12)(9,14,16,13)(17,19,22,27)(18,23,24,30)(20,25,28,31)(21,26,29,32)(33,35)(34,40)(36,47)(37,48)(38,43)(39,46)(41,45)(42,44);(18,24)(20,28)(23,30)(25,31)(34,41)(35,46)(36,43)(44,48);(1,4,7,3)(2,8,10,5)(6,12,15,11)(9,14,16,13)(17,20)(18,21)(19,25)(22,28)(23,26)(24,29)(27,31)(30,32)(33,36,39,43)(34,42,41,37)(35,47,46,38)(40,44,45,48);(1,5,7,8)(2,3,10,4)(6,14,15,13)(9,12,16,11)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,37,39,42)(34,43,41,36)(35,44,46,48)(38,40,47,45);(1,6,7,15)(2,9,10,16)(3,11,4,12)(5,13,8,14)(17,21)(18,20)(19,26)(22,29)(23,25)(24,28)(27,32)(30,31)(33,38,39,47)(34,44,41,48)(35,43,46,36)(37,40,42,45)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817251);