#284951

order	512
nilpotency class	3
isoclinism	isoclinic class of #284923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 192147);