#395950

order	512
nilpotency class	3
isoclinism	isoclinic class of #395950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14729
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1345 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303146);