#317680

order	512
nilpotency class	3
isoclinism	isoclinic class of #262731
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=gh,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224876);