#8772669

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4008 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=h,e^2=gh,f^2=h,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, 8679865);