#299264

order	512
nilpotency class	3
isoclinism	isoclinic class of #299253
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18624	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

992 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)(3,4,8,6)(9,10,16,20)(11,13,23,24)(12,14,19,15)(17,22,21,18);(2,7)(3,8)(9,11,15,22,16,23,14,18)(10,17,19,24,20,21,12,13);(1,3,6,2,5,8,4,7)(9,12,15,10,16,19,14,20)(11,21,22,13,23,17,18,24);(1,4,5,6)(2,3,7,8)(9,13,14,21,16,24,15,17)(10,18,12,23,20,22,19,11)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=f,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,g^b=gh,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206460);