#181107

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1551 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=i,g^2=hi,b^a=be,c^a=ci,e^a=eh,c^b=ci,f^b=fh,d^c=dh

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, 88303);