#4435295

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×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

2633 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=fg,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,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, 4342491);