#336278

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25392	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ehi,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 243474);