#9718573

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	#51687	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4139 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,e^2=g,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, 9625769);