#9718591

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
prime radical	#51846	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 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)(3,4)(5,13)(6,7)(8,10)(9,15)(11,14)(12,16)(17,18,22,24)(19,25,28,23)(20,21,29,31)(26,30,32,27)(33,34)(35,43)(36,45)(37,42)(38,46)(39,41)(40,44)(47,48);(1,3)(2,4)(5,11)(6,10)(7,8)(9,12)(13,14)(15,16)(17,19,22,28)(18,23,24,25)(20,26,29,32)(21,27,31,30)(33,35,39,44)(34,40,41,43)(36,37,46,47)(38,42,45,48);(2,7)(3,10)(9,13)(11,16)(18,24)(20,29)(23,25)(26,32)(34,41)(35,44)(36,46)(42,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,14)(12,13)(15,16)(17,20)(18,21)(19,26)(22,29)(23,27)(24,31)(25,30)(28,32)(33,36)(34,38)(35,37)(39,46)(40,42)(41,45)(43,48)(44,47);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,19)(18,25)(20,26)(21,30)(22,28)(23,24)(27,31)(29,32)(33,37)(34,42)(35,36)(38,40)(39,47)(41,48)(43,45)(44,46);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,21)(18,20)(19,27)(22,31)(23,26)(24,29)(25,32)(28,30)(33,38)(34,36)(35,42)(37,40)(39,45)(41,46)(43,47)(44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=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, 9625787);