#7726845

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	#51700	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4313 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=gh,e^2=hi,f^2=g,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, 7634041);