#8818171

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

3865 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725367);