#7852767

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4501 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)(9,10,14,17)(11,18,15,12)(13,16,21,23)(19,24,22,20)(25,26)(27,33)(28,35)(29,31)(30,32)(34,37)(36,39)(38,40);(1,2,5,6)(3,7,4,8)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,22)(23,24)(25,27)(26,28)(29,34)(30,35)(31,36)(32,33)(37,40)(38,39);(9,12,14,18)(10,15,17,11)(13,20,21,24)(16,22,23,19)(25,26,30,32)(27,33,35,28)(29,31,38,40)(34,37,39,36);(1,3,5,4)(2,7,6,8)(9,12)(10,11)(13,20)(14,18)(15,17)(16,19)(21,24)(22,23)(25,26)(27,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40);(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,22)(23,24)(25,28)(26,27)(29,37)(30,33)(31,39)(32,35)(34,40)(36,38);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,16,17,23)(11,19,15,22)(12,20,18,24)(25,29,30,38)(26,31,32,40)(27,34,35,39)(28,36,33,37)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7759963);