#8167458

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	#42104	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=gi,f^2=h,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, 8074654);