#186195

order	512
nilpotency class	3
isoclinism	isoclinic class of #186128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3522	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2413 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93391);