#8924719

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	#57609	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3934 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)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(33,34)(35,42)(36,37)(38,39)(40,45)(41,46)(43,48)(44,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,23,26)(19,27,21,24)(20,22,25,32)(28,31,30,29)(34,39)(36,41)(40,42)(43,47);(2,6)(4,13)(7,9)(10,16)(17,19,23,21)(18,24,26,27)(20,30,25,28)(22,31,32,29)(33,34,38,39)(35,40,45,42)(36,37,41,46)(43,44,47,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20,23,25)(18,22,26,32)(19,28,21,30)(24,29,27,31)(33,35,38,45)(34,40,39,42)(36,43,41,47)(37,44,46,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21,23,19)(18,24,26,27)(20,28,25,30)(22,31,32,29)(33,36,38,41)(34,37,39,46)(35,43,45,47)(40,44,42,48);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,22)(18,25)(19,29)(20,26)(21,31)(23,32)(24,30)(27,28)(33,37)(34,41)(35,44)(36,39)(38,46)(40,47)(42,43)(45,48)

Polycyclic presentation:

2^9|b^2=hi,c^2=ghi,d^2=ghi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,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, 8831915);