#235401

order	512
nilpotency class	3
isoclinism	isoclinic class of #235396
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

1207 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)(3,13)(4,14)(5,10)(6,11)(7,12)(8,15)(9,16)(17,18)(19,28)(20,25)(21,26)(22,31)(23,27)(24,30)(29,32);(1,3,7,9)(2,8,12,4)(5,14,6,15)(10,13,11,16)(17,19,23,20)(18,24,27,32)(21,30,22,29)(25,31,28,26);(1,4,7,8)(2,9,12,3)(5,16,6,13)(10,15,11,14)(17,20,23,19)(18,25,27,28)(21,29,22,30)(24,26,32,31);(1,5,7,6)(2,10,12,11)(3,14,9,15)(4,16,8,13)(17,21,23,22)(18,26,27,31)(19,29,20,30)(24,28,32,25)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=di,e^b=eh,d^c=di,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, 142597);