#8696222

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	#2828	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3945 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603418);