#415362

order	512
nilpotency class	3
isoclinism	isoclinic class of #394267
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Q₈×#2029
maximal normal subgroup of	Q₈×#2029
maximal quotient of	#72267901, #72267902

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322558);