#251757

order	512
nilpotency class	3
isoclinism	isoclinic class of #251731
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18304	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

825 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)(3,12)(4,11)(5,10)(6,9)(7,14)(8,13)(15,16)(18,23)(20,21)(24,26)(25,31)(28,29)(30,32);(1,3)(2,7)(4,15)(5,14)(6,13)(8,11)(9,16)(10,12)(18,24,26,23)(19,27)(25,30,32,31)(28,29);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,12)(10,16)(17,18)(19,25)(20,24)(21,23)(22,26)(27,32)(28,30)(29,31);(1,4,5,6)(2,9,10,11)(3,13,14,15)(7,8,12,16)(17,19,22,27)(18,25,26,32)(20,29,21,28)(23,30,24,31)

Polycyclic presentation:

2^9|b^2=ehi,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^c=eh,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158953);