#479854

order	512
nilpotency class	3
isoclinism	isoclinic class of #478457
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

858 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,20,22,15)(13,16)(14,19)(18,23,21,24);(2,5)(6,7)(9,11)(10,14,16,17)(12,21,22,18)(13,19)(15,24)(20,23);(1,3,4,8)(2,6,5,7)(18,21)(23,24);(3,8)(6,7)(9,12)(10,15)(11,18)(13,22)(14,23)(16,20)(17,24)(19,21)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387050);