#263100

order	512
nilpotency class	3
isoclinism	isoclinic class of #263075
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

891 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)(3,8)(4,7)(5,6)(9,10,14,11)(12,20,22,24)(13,18,15,17)(16,23,19,21);(1,3,4,5)(2,6,7,8)(9,11,13,18,14,10,15,17)(12,19,21,24,22,16,23,20);(2,6)(3,5)(7,8)(9,10)(11,14)(12,19)(13,17)(15,18)(16,22)(20,23)(21,24);(2,7)(6,8)(9,12,13,21,14,22,15,23)(10,16,17,20,11,19,18,24)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,g^b=gh,d^c=dh,f^c=fi,g^c=gh

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, 170296);