#194386

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60900	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1081 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101582);