#7732475

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	#51171	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4217 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=gh,e^2=i,f^2=hi,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, 7639671);