#268049

order	512
nilpotency class	3
isoclinism	isoclinic class of #268043
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

885 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:

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

Polycyclic presentation:

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

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, 175245);