#9814647

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	#40656	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=gi,e^2=g,f^2=gh,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, 9721843);