#3465349

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2228 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)(9,10)(11,16)(12,21)(13,17)(14,15)(18,19)(20,23)(22,24)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(5,6,7,8)(10,15)(12,17)(16,23)(18,24)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(5,6)(7,8)(9,11)(10,16)(12,18)(13,22)(14,20)(15,23)(17,24)(19,21)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,2,3,4)(9,12)(10,13)(11,18)(14,17)(15,21)(16,22)(19,23)(20,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(9,13)(10,17)(11,19)(12,15)(14,21)(16,18)(20,22)(23,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3372545);