#7503352

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3525 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:

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

Polycyclic presentation:

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

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, 7410548);