#8094969

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

4130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002165);