#9712843

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3925 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,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, 9620039);