#7711538

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	#45800	C₂
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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618734);