#348674

order	512
nilpotency class	3
isoclinism	isoclinic class of #348674
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#18135	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1084 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:

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

Polycyclic presentation:

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

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, 255870);