#245639

order	512
nilpotency class	3
isoclinism	isoclinic class of #245615
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1382 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152835);