#9714625

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	#36906	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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)(19,23)(20,26)(21,29)(22,24)(25,28)(27,32)(30,31)(33,34)(35,42)(36,41)(37,39)(38,40)(43,48)(44,46)(45,47);(3,9)(7,13)(10,15)(14,16)(17,19)(18,23)(20,27)(21,30)(22,28)(24,25)(26,32)(29,31)(33,35,38,37)(34,39,40,42)(36,44,45,43)(41,48,47,46);(2,6)(7,13)(8,11)(14,16)(18,24)(19,28)(26,30)(27,29)(34,40)(39,42)(41,47)(46,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,25)(20,29)(21,30)(22,28)(23,24)(26,32)(27,31)(33,34)(35,39)(36,41)(37,42)(38,40)(43,46)(44,48)(45,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20)(18,26)(19,29)(21,23)(22,31)(24,30)(25,32)(27,28)(33,36,38,45)(34,41,40,47)(35,43,37,44)(39,46,42,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,32)(18,27,24,29)(19,30,28,26)(20,25,31,23)(33,37,38,35)(34,42,40,39)(36,43,45,44)(41,46,47,48)

Polycyclic presentation:

2^9|b^2=g,e^2=g,f^2=gi,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, 9621821);