#497660

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×Q₈² ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

867 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=ei,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271393
maximal normal subgroup of	#72271393
maximal quotient of	#72271424

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404856);