#235368

order	512
nilpotency class	3
isoclinism	isoclinic class of #235354
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
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1329 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)(3,13)(4,16)(5,10)(6,11)(7,12)(8,15)(9,14)(17,18)(19,27)(20,24)(21,30)(22,26)(23,28)(25,31)(29,32);(1,3,7,4)(2,8,12,9)(5,14,6,15)(10,13,11,16)(17,19,22,20)(18,23,26,32)(21,29,31,28)(24,30,27,25);(1,4,7,3)(2,9,12,8)(5,14,6,15)(10,13,11,16)(17,20,22,19)(18,24,26,27)(21,28,31,29)(23,25,32,30);(1,5,7,6)(2,10,12,11)(3,14,4,15)(8,13,9,16)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142564);