#295209

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#30265	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

876 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,25)(20,29)(21,24)(22,31)(23,27)(26,28)(30,32);(2,5)(6,7)(9,10,13,15)(11,14,16,12)(17,19,21,27)(18,23,24,25)(20,30,31,26)(22,32,29,28);(9,11,12,15,13,16,14,10)(17,20,29,18,21,31,22,24)(19,26,32,23,27,30,28,25);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,11,15,16)(18,24)(20,31)(23,25)(26,30)

Polycyclic presentation:

2^9|b^2=h,c^2=fh,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=di,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 202405);