#318606

order	512
nilpotency class	3
isoclinism	isoclinic class of #318601
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#28240	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

604 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:

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

Polycyclic presentation:

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

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, 225802);