#285015

order	512
nilpotency class	3
isoclinism	isoclinic class of #284987
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

636 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:

(1,2,6,9,7,4,5,14)(3,10,12,8,13,16,15,11)(17,18,21,27,23,26,22,28)(19,24,29,25,20,31,30,32);(1,3,5,15,7,13,6,12)(2,8,14,10,4,11,9,16)(17,19,21,29,23,20,22,30)(18,24,27,25,26,31,28,32);(1,4)(2,7)(3,11)(5,14)(6,9)(8,13)(10,15)(12,16)(17,20,23,19)(18,25,26,32)(21,29,22,30)(24,28,31,27);(1,2)(3,8)(4,7)(5,14)(6,9)(10,15)(11,13)(12,16)(18,26)(24,31)(25,32)(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=dei,e^a=eh,c^b=cg,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, 192211);