#9812861

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	#54942	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4305 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=gi,e^2=g,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, 9720057);