#8588359

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	#38510	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4284 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)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31)(33,34,39,41)(35,43,45,40)(36,44,47,42)(37,46,48,38);(1,3,7,9)(2,4,5,12)(6,10,15,16)(8,11,13,14)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,35,39,45)(34,40,41,43)(36,46,47,38)(37,42,48,44);(2,5)(3,9)(8,13)(10,16)(21,30)(24,28)(26,29)(31,32)(34,41)(37,48)(40,43)(42,44);(1,4)(2,3)(5,9)(6,14)(7,12)(8,16)(10,13)(11,15)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,36,39,47)(34,42,41,44)(35,38,45,46)(37,43,48,40);(1,5,7,2)(3,4,9,12)(6,13,15,8)(10,11,16,14)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,37)(34,38)(35,44)(36,43)(39,48)(40,47)(41,46)(42,45);(1,6)(2,8)(3,10)(4,11)(5,13)(7,15)(9,16)(12,14)(17,19)(18,25)(20,23)(21,26)(22,27)(24,32)(28,31)(29,30)(33,38)(34,37)(35,36)(39,46)(40,42)(41,48)(43,44)(45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8495555);