#7731102

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	#37845	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4419 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7638298);