#283720

order	512
nilpotency class	3
isoclinism	isoclinic class of #283707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3035	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

661 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=hi,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=gi,f^c=fh,g^c=ghi,e^d=eh

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, 190916);