#8664350

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	#49808	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571546);