#9712416

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	#45680	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3987 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)(17,18,22,24)(19,23,26,20)(21,29,31,25)(27,28,32,30)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(4,11)(8,14)(10,15)(12,16)(17,19)(18,23)(20,24)(21,27)(22,26)(25,30)(28,29)(31,32)(33,35)(34,36)(37,45)(38,39)(40,47)(41,42)(43,46)(44,48);(2,7)(3,9)(8,14)(10,15)(18,24)(19,26)(25,29)(27,32)(33,36,38,42)(34,39,41,35)(37,44,46,47)(40,45,48,43);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,30)(31,32)(33,36)(34,35)(37,44)(38,42)(39,41)(40,43)(45,48)(46,47);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,20)(18,19)(21,30)(22,23)(24,26)(25,32)(27,29)(28,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,45)(41,48)(42,47);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,21,22,31)(18,25,24,29)(19,27,26,32)(20,28,23,30)(33,35)(34,36)(37,45)(38,39)(40,47)(41,42)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=h,c^2=g,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9619612);