#264822

order	512
nilpotency class	3
isoclinism	isoclinic class of #264819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18678	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

846 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dh,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, 172018);