#9847638

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	#51967	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) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9754834);