#185271

order	512
nilpotency class	3
isoclinism	isoclinic class of #185267
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	#62819	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2482 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,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, 92467);