#270800

order	512
nilpotency class	3
isoclinism	isoclinic class of #270747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3498	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

945 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=gi,f^c=fh,g^c=gi,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, 177996);