#265203

order	512
nilpotency class	3
isoclinism	isoclinic class of #265163
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

979 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)(3,4)(5,7)(6,8)(9,10)(11,18)(12,15)(13,23)(14,17)(16,20)(19,24)(21,22);(1,3,6,7)(2,4,8,5)(9,10,15,23,14,17,13,12)(11,16,20,18,21,24,19,22);(2,3)(4,5)(7,8)(9,11)(10,16)(12,22)(13,19)(14,21)(15,20)(17,24)(18,23);(1,4,6,5)(2,3,8,7)(9,12,15,10,14,23,13,17)(11,18,20,24,21,22,19,16)

Polycyclic presentation:

2^9|b^2=ei,d^2=ei,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=ghi,c^b=cg,d^b=dh,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172399);