#285417

order	512
nilpotency class	3
isoclinism	isoclinic class of #285371
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	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

574 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192613);