#267526

order	512
nilpotency class	3
isoclinism	isoclinic class of #267467
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

828 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 174722);