#8695888

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	#50011	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4255 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^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, 8603084);