#295969

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

713 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=h,c^2=f,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 203165);