#295069

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

754 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)(3,7)(4,5)(6,8)(9,10,13,15)(11,12,16,14)(17,18)(19,21)(20,27)(22,26)(23,29)(24,30)(25,31)(28,32);(2,5)(6,7)(9,11,13,16)(10,14,15,12)(17,19,21,18)(20,25,28,23)(22,26,30,24)(27,29,32,31);(9,11,14,10,13,16,12,15)(17,20,22,29,21,28,30,31)(18,23,24,27,19,25,26,32);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,11,15,16)(18,19)(20,28)(24,26)(29,31)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi

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, 202265);