#220230

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	C₂×#577	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1053 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,d^b=dei,e^b=eh,f^c=fh,g^c=gh,e^d=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127426);