#7709302

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	#45788	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4464 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:

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

Polycyclic presentation:

2^9|b^2=hi,d^2=gh,e^2=g,f^2=g,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, 7616498);