#263768

order	512
nilpotency class	3
isoclinism	isoclinic class of #263759
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3559	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

768 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170964);