#7514301

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

3186 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^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,d^c=di,f^e=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421497);