#185543

order	512
nilpotency class	3
isoclinism	isoclinic class of #185463
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3505	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2467 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,e^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=ghi,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, 92739);