#9718229

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	#37171	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,23)(19,24,27,31)(20,25,30,29)(21,28,32,26)(33,34,38,39)(35,37,41,47)(36,40,45,48)(42,46,43,44);(3,9)(7,13)(10,15)(14,16)(18,23)(20,30)(24,31)(26,28)(33,35)(34,37)(36,43)(38,41)(39,47)(40,44)(42,45)(46,48);(2,6)(7,13)(8,11)(14,16)(18,23)(19,27)(25,29)(26,28)(34,39)(35,41)(40,48)(42,43);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,24)(20,28)(21,29)(22,27)(23,31)(25,32)(26,30)(33,34)(35,37)(36,40)(38,39)(41,47)(42,46)(43,44)(45,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20)(18,25)(19,28)(21,31)(22,30)(23,29)(24,32)(26,27)(33,36,38,45)(34,40,39,48)(35,42,41,43)(37,46,47,44);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,32)(18,26,23,28)(19,29,27,25)(20,24,30,31)(33,37)(34,35)(36,44)(38,47)(39,41)(40,43)(42,48)(45,46)

Polycyclic presentation:

2^9|a^2=gi,e^2=gh,f^2=g,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, 9625425);