#9723155

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	#55697	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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)(3,12)(4,7)(5,10)(6,9)(8,11)(13,16)(14,15)(17,18,23,28)(19,24,29,32)(20,30,31,25)(21,26,27,22)(33,34,39,37)(35,42,44,40)(36,45,38,41)(43,47,46,48);(1,3,8,6)(2,9,11,12)(4,13,5,14)(7,15,10,16)(17,19,23,29)(18,24,28,32)(20,26,31,22)(21,30,27,25)(33,35)(34,40)(36,43)(37,42)(38,46)(39,44)(41,47)(45,48);(1,4,8,5)(2,7,11,10)(3,13,6,14)(9,15,12,16)(17,20,23,31)(18,25,28,30)(19,22,29,26)(21,24,27,32)(33,36,39,38)(34,41,37,45)(35,43,44,46)(40,47,42,48);(1,5,8,4)(2,10,11,7)(3,14,6,13)(9,16,12,15)(17,20)(18,25)(19,26)(21,24)(22,29)(23,31)(27,32)(28,30)(33,37,39,34)(35,42,44,40)(36,41,38,45)(43,47,46,48);(1,6,8,3)(2,9,11,12)(4,14,5,13)(7,15,10,16)(17,21)(18,26)(19,25)(20,24)(22,28)(23,27)(29,30)(31,32)(33,38,39,36)(34,41,37,45)(35,43,44,46)(40,48,42,47);(1,7,8,10)(2,4,11,5)(3,15,6,16)(9,13,12,14)(17,22,23,26)(18,27,28,21)(19,20,29,31)(24,25,32,30)(33,35)(34,40)(36,43)(37,42)(38,46)(39,44)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=g,f^2=gh,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, 9630351);