#9727196

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	#57649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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)(19,26)(21,23)(27,32)(29,31)(33,34)(35,43)(36,40)(37,47)(38,42)(39,44)(41,45)(46,48);(3,9)(7,13)(10,15)(14,16)(17,18,22,25)(19,23,26,21)(20,28,30,24)(27,31,32,29)(33,35,38,44)(34,39,42,43)(36,45,46,37)(40,47,48,41);(2,6)(7,13)(8,11)(14,16)(17,19,22,26)(18,21,25,23)(20,27,30,32)(24,31,28,29)(33,34,38,42)(35,39,44,43)(36,40,46,48)(37,41,45,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,29)(28,31)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,44)(45,47)(46,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,30)(18,24,25,28)(19,27,26,32)(21,31,23,29)(33,36,38,46)(34,40,42,48)(35,37,44,45)(39,41,43,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,23)(18,19,25,26)(20,29,30,31)(24,32,28,27)(33,37)(34,41)(35,36)(38,45)(39,40)(42,47)(43,48)(44,46)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,e^2=h,f^2=i,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, 9634392);