#253623

order	512
nilpotency class	3
isoclinism	isoclinic class of #253603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160819);