#9814965

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	#52329	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3962 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(34,40)(36,45)(41,46)(43,44);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(19,25)(23,26)(27,29)(31,32)(33,34)(35,41)(36,42)(37,46)(38,43)(39,40)(44,48)(45,47);(2,3)(7,9)(8,10)(14,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,35,39,37)(34,41,40,46)(36,44,45,43)(38,42,48,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,22,18)(19,26,25,23)(21,28,30,24)(27,32,29,31)(33,36)(34,42)(35,44)(37,43)(38,46)(39,45)(40,47)(41,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31)(33,37,39,35)(34,41,40,46)(36,43,45,44)(38,42,48,47);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,48)(34,43,40,44)(35,42,37,47)(36,46,45,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=i,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, 9722161);