#8430418

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	#2846	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4803 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8337614);