#8994423

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3850 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8901619);