#9712797

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

3914 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)(11,16)(13,15)(19,22)(20,24)(27,31)(28,32)(33,34,40,43)(35,44,37,38)(36,45,46,41)(39,47,48,42);(1,2)(3,8)(4,7)(5,9)(6,10)(11,15)(12,14)(13,16)(17,18)(19,20)(21,29)(22,24)(23,26)(25,30)(27,32)(28,31)(33,35,40,37)(34,38,43,44)(36,42,46,47)(39,45,48,41);(1,3)(2,5)(4,11)(6,9)(7,13)(8,10)(12,16)(14,15)(17,19,23,22)(18,24,26,20)(21,27,30,31)(25,32,29,28)(33,36,40,46)(34,41,43,45)(35,42,37,47)(38,39,44,48);(1,4)(2,7)(3,11)(5,13)(6,12)(8,15)(9,16)(10,14)(17,20,23,24)(18,19,26,22)(21,28,30,32)(25,27,29,31)(33,37,40,35)(34,38,43,44)(36,42,46,47)(39,41,48,45);(1,3)(2,8)(4,11)(5,10)(6,9)(7,15)(12,16)(13,14)(17,21)(18,25)(19,27)(20,28)(22,31)(23,30)(24,32)(26,29)(33,38)(34,35)(36,39)(37,43)(40,44)(41,42)(45,47)(46,48);(1,5,6,8)(2,9,10,3)(4,13,12,15)(7,16,14,11)(17,22,23,19)(18,24,26,20)(21,27,30,31)(25,28,29,32)(33,39)(34,42)(35,45)(36,44)(37,41)(38,46)(40,48)(43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^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, 9619993);