#235309

order	512
nilpotency class	3
isoclinism	isoclinic class of #235303
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1387 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)(3,7)(4,12)(5,9)(6,11)(8,14)(10,16)(13,15)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,31)(30,32);(1,3)(2,6)(4,13)(5,10)(7,12)(8,16)(9,15)(11,14)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,27)(26,31);(1,3)(2,7)(4,11)(5,10)(6,12)(8,15)(9,16)(13,14)(18,24)(20,21)(23,27)(25,26)(28,29)(31,32);(3,10)(6,15)(7,16)(11,13)(17,20,22,21)(18,25,26,24)(19,28,30,29)(23,27,31,32)

Polycyclic presentation:

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

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, 142505);