#8135663

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	#56376	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4188 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8042859);