#7514291

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
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 ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3361 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(5,6)(7,8)(9,10,15,19)(11,20,13,23)(12,18,14,17)(16,22,24,21);(1,2)(3,4)(5,6)(7,8)(9,11,15,13)(10,16,19,24)(12,21,14,22)(17,20,18,23);(6,8)(9,12,15,14)(10,17,19,18)(11,21,13,22)(16,20,24,23);(2,4)(6,8)(9,12,15,14)(10,17,19,18)(11,21,13,22)(16,20,24,23);(9,13,15,11)(10,16,19,24)(12,22,14,21)(17,20,18,23)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,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, 7421487);