#237538

order	512
nilpotency class	3
isoclinism	isoclinic class of #237527
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

1218 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)(4,12)(5,8)(6,9)(7,10)(13,14)(15,16)(17,18,21,26)(19,27,29,31)(20,24,22,25)(23,30,32,28);(1,3,7,13)(2,4,10,15)(5,16,6,12)(8,11,9,14)(17,19,21,29)(18,23,26,32)(20,30,22,28)(24,27,25,31);(1,4)(2,3)(5,11)(6,14)(7,15)(8,16)(9,12)(10,13)(17,18)(19,27)(20,24)(21,26)(22,25)(23,28)(29,31)(30,32);(1,5,7,6)(2,8,10,9)(3,12,13,16)(4,14,15,11)(17,20,21,22)(18,24,26,25)(19,28,29,30)(23,31,32,27)

Polycyclic presentation:

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

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, 144734);