#323174

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	#28497	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

741 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)(9,10,16,19,15,12,14,20)(11,21,24,23,22,18,13,17);(1,3,4,7,5,2,6,8)(9,11,16,24,15,22,14,13)(10,17,20,18,12,23,19,21);(1,2,5,3)(4,7,6,8)(9,12,15,10)(11,17,22,23)(13,21,24,18)(14,20,16,19);(1,4,5,6)(2,7,3,8)(9,13,16,11,15,24,14,22)(10,18,20,23,12,21,19,17)

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,c^2=i,d^2=eh,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, 230370);