#9712178

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	#43182	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18,23,26)(19,21,25,31)(20,22,30,24)(27,32,28,29)(34,36)(35,43)(38,48)(41,46);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19,23,25)(18,21,26,31)(20,28,30,27)(22,29,24,32)(33,34,39,36)(35,40,43,45)(37,46,44,41)(38,42,48,47);(2,4)(3,10)(9,12)(11,16)(17,20,23,30)(18,24,26,22)(19,27,25,28)(21,29,31,32)(33,35,39,43)(34,40,36,45)(37,38,44,48)(41,47,46,42);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20)(18,22)(19,28)(21,29)(23,30)(24,26)(25,27)(31,32)(33,36,39,34)(35,40,43,45)(37,46,44,41)(38,47,48,42);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,21,23,31)(18,25,26,19)(20,29,30,32)(22,27,24,28)(33,37)(34,41)(35,38)(36,46)(39,44)(40,47)(42,45)(43,48);(1,6,7,14)(2,9,4,12)(3,11,10,16)(5,13,8,15)(17,22)(18,20)(19,29)(21,28)(23,24)(25,32)(26,30)(27,31)(33,38)(34,42)(35,44)(36,47)(37,43)(39,48)(40,41)(45,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=g,e^2=gh,f^2=ghi,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, 9619374);