#4100327

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 2097152
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1695 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fgh,c^2=hi,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4007523);