#9717462

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	#48464	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=gi,f^2=h,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, 9624658);