#8551004

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=h,e^2=h,f^2=gi,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, 8458200);