#7516558

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61075	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1925 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)(3,6)(4,5)(7,8)(9,10)(11,12)(13,27)(14,19)(15,21)(16,23)(17,26)(18,30)(20,31)(22,38)(24,32)(25,35)(28,40)(29,37)(33,36)(34,39);(2,5)(6,8)(9,11,14,23)(10,16,19,12)(13,28,29,22)(15,17,31,24)(18,34,36,25)(20,26,21,32)(27,38,37,40)(30,35,33,39);(9,12)(10,17)(11,21)(13,25)(14,16)(15,32)(18,35)(19,24)(20,23)(22,27)(26,31)(28,37)(29,34)(30,40)(33,38)(36,39);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,22)(12,25)(14,29)(15,33)(16,34)(17,35)(19,36)(20,37)(21,27)(23,28)(24,39)(26,40)(30,31)(32,38);(2,5)(6,8)(10,19)(12,16)(13,29)(20,21)(22,28)(26,32)(30,33)(35,39)

Polycyclic presentation:

2^9|b^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7423754);