#245423

order	512
nilpotency class	3
isoclinism	isoclinic class of #245375
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	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1281 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:

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

Polycyclic presentation:

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

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, 152619);