#185423

order	512
nilpotency class	3
isoclinism	isoclinic class of #185267
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	#62830	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2247 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(2,6)(4,11)(7,15)(8,12)(10,13)(14,16)(17,18,21,23,22,26,20,25)(19,24,29,30,27,31,28,32);(1,2)(3,7)(4,12)(5,8)(6,11)(9,16)(10,14)(13,15)(18,23)(20,21)(24,30)(25,26)(28,29)(31,32);(1,3)(2,7)(4,13)(5,9)(6,14)(8,16)(10,11)(12,15)(17,19)(18,24)(20,28)(21,29)(22,27)(23,30)(25,32)(26,31);(3,9)(7,16)(10,13)(14,15)(19,27)(24,31)(28,29)(30,32)

Polycyclic presentation:

2^9|a^2=g,e^2=i,g^2=hi,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, 92619);