#557678

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 464874);