#8889059

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3863 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8796255);