#8535055

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	#58010	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4155 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8442251);