#9937709

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	#41827	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9844905);