#185274

order	512
nilpotency class	3
isoclinism	isoclinic class of #185273
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62824	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92470);