#8108723

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	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34,38,39)(35,42,37,40)(36,44,45,41)(43,47,46,48);(2,6)(7,9)(8,12)(14,15)(18,22)(23,25)(24,29)(30,31)(34,39)(36,45)(40,42)(43,46);(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,38,37)(34,40,39,42)(36,43,45,46)(41,47,44,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,45)(34,41,39,44)(35,43,37,46)(40,47,42,48);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,38,35)(34,42,39,40)(36,46,45,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,e^2=h,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^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015919);