#185290

order	512
nilpotency class	3
isoclinism	isoclinic class of #185285
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	#62826	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92486);