#544280

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2222 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:

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

Polycyclic presentation:

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

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, 451476);