#9712119

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3805 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=g,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, 9619315);