#7514030

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3172 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:

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

Polycyclic presentation:

2^9|e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,f^b=fi,d^c=di,f^e=fi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421226);