#263780

order	512
nilpotency class	3
isoclinism	isoclinic class of #263779
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

963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

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

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, 170976);