#7782705

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	#45353	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4542 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)(11,12)(13,18)(14,16)(15,17)(19,20)(21,23)(22,24)(25,26)(27,38)(28,34)(29,35)(30,33)(31,36)(32,37)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,16)(13,19)(14,17)(18,22)(20,23)(21,24)(25,27,32,30)(26,33,37,38)(28,36,29,39)(31,34,40,35);(10,16)(11,17)(13,21)(20,22)(25,28,32,29)(26,34,37,35)(27,39,30,36)(31,38,40,33);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,24)(22,23)(25,29,32,28)(26,35,37,34)(27,39,30,36)(31,38,40,33);(1,4,5,3)(2,8,6,7)(9,12)(10,17)(11,16)(13,20)(14,15)(18,24)(19,23)(21,22)(25,30,32,27)(26,33,37,38)(28,36,29,39)(31,35,40,34);(9,13,14,21)(10,18,16,23)(11,19,17,24)(12,20,15,22)(25,31,32,40)(26,36,37,39)(27,34,30,35)(28,33,29,38)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=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, 7689901);