#9727004

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,25)(19,28,30,24)(20,26,31,21)(22,29,32,27)(35,42)(37,39)(43,48)(45,47);(3,9)(7,13)(10,15)(14,16)(17,19,23,30)(18,24,25,28)(20,29,31,27)(21,32,26,22)(33,34,38,41)(35,39,42,37)(36,44,46,40)(43,47,48,45);(2,6)(7,13)(8,11)(14,16)(18,25)(20,31)(24,28)(27,29)(33,35,38,42)(34,37,41,39)(36,43,46,48)(40,47,44,45);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,26)(19,29)(21,25)(22,24)(23,31)(27,30)(28,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,45)(44,47)(46,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,26)(18,20,25,31)(19,22,30,32)(24,27,28,29)(33,36,38,46)(34,40,41,44)(35,43,42,48)(37,47,39,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22)(18,27)(19,21)(20,24)(23,32)(25,29)(26,30)(28,31)(33,37,38,39)(34,35,41,42)(36,45,46,47)(40,48,44,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,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, 9634200);