#9851420

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	#48564	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,23,26)(19,27,29,24)(20,30,31,22)(21,25,32,28)(33,34,37,41)(35,40,43,38)(36,44,45,39)(42,46,48,47);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,23,29)(18,24,26,27)(20,28,31,25)(21,30,32,22)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(2,7)(3,10)(8,15)(11,12)(33,36,37,45)(34,39,41,44)(35,42,43,48)(38,46,40,47);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20)(18,22)(19,28)(21,24)(23,31)(25,29)(26,30)(27,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,47)(41,44)(43,48);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,21,23,32)(18,25,26,28)(19,22,29,30)(20,24,31,27)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,46)(45,48);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,22)(18,20)(19,21)(23,30)(24,28)(25,27)(26,31)(29,32)(33,34)(35,38)(36,39)(37,41)(40,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9758616);