#251712

order	512
nilpotency class	3
isoclinism	isoclinic class of #251701
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

807 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,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^c=di,e^c=eh,f^c=fi,g^c=gh,f^d=fh

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, 158908);