#275870

order	512
nilpotency class	3
isoclinism	isoclinic class of #267051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

693 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 183066);