#357041

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1024 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264237);