#8174611

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4185 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:

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

Polycyclic presentation:

2^9|c^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8081807);