#8167441

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	#57217	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8074637);