#9712303

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3803 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)(3,10)(4,7)(5,9)(6,8)(11,16)(12,14)(13,15)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,34,40,37)(35,39,42,44)(36,41,47,38)(43,45,46,48);(1,3,7,6)(2,8,4,10)(5,13,14,11)(9,16,12,15)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35)(34,39)(36,46)(37,44)(38,45)(40,42)(41,48)(43,47);(2,4)(8,10)(9,12)(15,16)(17,18,23,21)(19,24,20,25)(22,26,30,29)(27,31,28,32)(33,36,40,47)(34,41,37,38)(35,43,42,46)(39,45,44,48);(1,4,7,2)(3,10,6,8)(5,12,14,9)(11,16,13,15)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,37,40,34)(35,44,42,39)(36,38,47,41)(43,48,46,45);(1,5,7,14)(2,9,4,12)(3,11,6,13)(8,15,10,16)(17,21,23,18)(19,25,20,24)(22,26,30,29)(27,31,28,32)(33,38)(34,36)(35,45)(37,47)(39,46)(40,41)(42,48)(43,44);(1,6,7,3)(2,10,4,8)(5,11,14,13)(9,15,12,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,39,40,44)(34,42,37,35)(36,45,47,48)(38,43,41,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,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, 9619499);