#4416783

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 8388608
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

2560 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)(10,11)(12,19)(13,21)(22,23)(25,26,30,33)(27,28,35,31)(29,32,38,40)(34,36,37,39);(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,27,30,35)(26,31,33,28)(29,34,38,37)(32,39,40,36);(5,6,8,7)(9,11,14,10)(12,17,19,15)(13,16,21,18)(20,22,24,23)(25,27)(26,28)(29,37)(30,35)(31,33)(32,39)(34,38)(36,40);(1,2,3,4)(5,7,8,6)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=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, 4323979);