#285543

order	512
nilpotency class	3
isoclinism	isoclinic class of #285499
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	#3106	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

591 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)(4,11,12,16)(5,7,15,9)(8,14,10,13)(17,18,21,30,23,20,22,26)(19,27,31,25,29,32,28,24);(2,7)(3,9)(4,12)(5,15)(8,16)(10,11)(17,19,22,28,23,29,21,31)(18,24,26,32,20,25,30,27);(1,3,6,2)(4,10,12,8)(5,7,15,9)(11,13,16,14)(17,19)(18,25)(20,24)(21,28)(22,31)(23,29)(26,32)(27,30);(1,4)(2,8)(3,10)(5,14)(6,12)(7,11)(9,16)(13,15)(17,20)(18,23)(19,24)(21,26)(22,30)(25,29)(27,31)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192739);