#9723169

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	#54970	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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)(3,12)(4,7)(5,10)(6,9)(8,11)(13,16)(14,15)(17,18,23,21)(19,24,29,31)(20,28,30,25)(22,27,32,26)(33,34,39,40)(35,43,38,41)(36,37,42,47)(44,45,46,48);(1,3,8,6)(2,9,11,12)(4,13,5,14)(7,15,10,16)(17,19,23,29)(18,24,21,31)(20,27,30,26)(22,28,32,25)(34,40)(37,47)(41,43)(45,48);(1,4,8,5)(2,7,11,10)(3,13,6,14)(9,15,12,16)(17,20,23,30)(18,25,21,28)(19,26,29,27)(22,24,32,31)(33,35,39,38)(34,41,40,43)(36,46,42,44)(37,45,47,48);(1,5,8,4)(2,10,11,7)(3,14,6,13)(9,16,12,15)(17,20)(18,25)(19,27)(21,28)(22,24)(23,30)(26,29)(31,32)(33,36,39,42)(34,37,40,47)(35,44,38,46)(41,48,43,45);(1,6,8,3)(2,9,11,12)(4,14,5,13)(7,15,10,16)(17,21,23,18)(19,24,29,31)(20,28,30,25)(22,26,32,27)(33,37)(34,42)(35,45)(36,40)(38,48)(39,47)(41,44)(43,46);(1,7,8,10)(2,4,11,5)(3,15,6,16)(9,13,12,14)(17,22)(18,26)(19,28)(20,24)(21,27)(23,32)(25,29)(30,31)(33,38,39,35)(34,43,40,41)(36,46,42,44)(37,45,47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=gh,f^2=g,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, 9630365);