#7502121

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23393	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 7409317);