#8692575

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=h,e^2=gh,f^2=h,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, 8599771);