#8472960

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	#43912	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4385 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8380156);