#254487

order	512
nilpotency class	3
isoclinism	isoclinic class of #254467
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	C₂³×C₄	C₂⁴
prime radical	#2431	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

926 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)(3,6)(4,7)(5,8)(9,14)(10,13)(11,15)(12,16)(17,18,21,22)(19,25,27,23)(20,24,29,30)(26,28,31,32);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,19)(18,23)(20,28,29,32)(21,27)(22,25)(24,31,30,26);(1,4,5,11)(2,7,8,15)(3,9,10,16)(6,12,13,14)(17,20,21,29)(18,24,22,30)(19,26,27,31)(23,28,25,32);(17,19)(18,23)(20,26)(21,27)(22,25)(24,28)(29,31)(30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,g^c=gh,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, 161683);