#469143

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1009 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fg,c^2=eg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,h^b=hi,d^c=de,h^c=hi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376339);