#766919

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=ghi,c^2=h,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 674115);