#7503355

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
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

3532 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)(9,10,13,18,14,19,29,30)(11,16,22,32,21,34,27,15)(12,23,25,36,26,37,28,17)(20,31,24,35,38,40,39,33);(5,6)(7,8)(9,11)(10,15)(12,24)(13,27)(14,21)(16,30)(17,35)(18,34)(19,32)(20,25)(22,29)(23,31)(26,39)(28,38)(33,36)(37,40);(2,4)(6,8)(9,11)(10,16)(12,20)(13,22)(14,21)(15,30)(17,33)(18,32)(19,34)(23,31)(24,25)(26,38)(27,29)(28,39)(35,36)(37,40);(2,4)(9,12,14,26)(10,17,19,36)(11,20,21,38)(13,28,29,25)(15,31,32,40)(16,33,34,35)(18,37,30,23)(22,39,27,24);(11,21)(15,32)(16,34)(20,38)(22,27)(24,39)(31,40)(33,35)

Polycyclic presentation:

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

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, 7410551);