#7720359

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	#38903	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4274 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)(10,16)(14,18)(17,20)(22,24)(25,26,30,33)(27,28,31,36)(29,37,38,32)(34,40,39,35);(1,3,5,8)(2,6,4,7)(9,10,15,16)(11,20,12,17)(13,14,19,18)(21,22,23,24)(25,27,30,31)(26,28,33,36)(29,34,38,39)(32,35,37,40);(1,2,5,4)(3,6,8,7)(9,11,15,12)(10,17,16,20)(13,23,19,21)(14,22,18,24)(25,26)(27,28)(29,32)(30,33)(31,36)(34,35)(37,38)(39,40);(1,2,5,4)(3,6,8,7)(9,12,15,11)(10,17,16,20)(13,23,19,21)(14,24,18,22)(25,28)(26,31)(27,33)(29,35)(30,36)(32,39)(34,37)(38,40);(1,4,5,2)(3,7,8,6)(9,13,15,19)(10,18,16,14)(11,21,12,23)(17,22,20,24)(25,29)(26,32)(27,34)(28,35)(30,38)(31,39)(33,37)(36,40);(9,14,15,18)(10,19,16,13)(11,22,12,24)(17,21,20,23)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7627555);