#7518921

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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

2751 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)(11,12)(13,23)(14,24)(15,18)(16,21)(17,22)(19,20);(1,2)(3,4)(5,6)(7,8)(9,11,20,13,15,17,14,21)(10,16,24,22,18,23,19,12);(6,8)(9,12)(10,17)(11,18)(13,19)(14,23)(15,22)(16,20)(21,24);(2,4)(6,8)(10,18)(12,22)(16,23)(19,24);(9,13,14,11,15,21,20,17)(10,12,19,23,18,22,24,16)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426117);