#8630813

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	#53531	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4183 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8538009);