#7709898

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	#45663	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4395 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)(9,10,15,12)(11,16,14,18)(13,20,22,17)(19,24,21,23)(26,31)(29,37)(32,38)(34,40);(1,2,4,5)(3,6,7,8)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,26)(27,32)(28,29)(30,31)(33,34)(35,38)(36,37)(39,40);(10,12)(16,18)(17,20)(23,24)(26,31)(28,36)(32,38)(33,39);(1,3,4,7)(2,6,5,8)(9,12,15,10)(11,18,14,16)(13,20,22,17)(19,24,21,23)(25,27,30,35)(26,32,31,38)(28,33,36,39)(29,34,37,40);(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,28)(26,29)(27,33)(30,36)(31,37)(32,34)(35,39)(38,40);(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,29)(26,28)(27,34)(30,37)(31,36)(32,33)(35,40)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7617094);