#468367

order	512
nilpotency class	3
isoclinism	isoclinic class of #454968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fi,h^c=hi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 375563);