#8127598

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	#56354	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4078 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=i,e^2=g,f^2=h,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^c=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, 8034794);