#8508859

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	#35044	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=gi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

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, 8416055);