#323193

order	512
nilpotency class	3
isoclinism	isoclinic class of #323133
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3151	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

728 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,c^2=hi,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^c=fi,g^c=ghi

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, 230389);