#348071

order	512
nilpotency class	3
isoclinism	isoclinic class of #348070
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18548	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

900 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,f^c=fi,g^c=gi,f^d=fi,g^d=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255267);