#254117

order	512
nilpotency class	3
isoclinism	isoclinic class of #254083
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	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

796 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161313);