#235149

order	512
nilpotency class	3
isoclinism	isoclinic class of #235139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3217	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142345);