#254217

order	512
nilpotency class	3
isoclinism	isoclinic class of #254179
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	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

849 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161413);