#253620

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	#25625	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

834 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=hi,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, 160816);