#283807

order	512
nilpotency class	3
isoclinism	isoclinic class of #283771
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	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

637 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=gi,d^c=di,f^c=fh,g^c=ghi,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, 191003);