#205581

order	512
nilpotency class	3
isoclinism	isoclinic class of #9755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1193 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=hi,e^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,g^b=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112777);