#495348

order	512
nilpotency class	4
isoclinism	isoclinic class of #2230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#476
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,10,17,28,16,27,34,11,18,26,13,9,14,8,15)(7,19,23,12,21,32,29,30,25,36,20,33,24,35,22,31);(1,2,3,4)(5,7,9,21,11,25,28,24)(6,12,16,31,18,33,14,30)(8,20,26,22,27,23,10,29)(13,19,15,35,17,36,34,32);(5,8,9,26,11,27,28,10)(6,13,16,15,18,17,14,34)(7,20,21,22,25,23,24,29)(12,19,31,35,33,36,30,32);(5,9,11,28)(6,14,18,16)(7,21,25,24)(8,26,27,10)(12,30,33,31)(13,34,17,15)(19,32,36,35)(20,22,23,29)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,d^2=i,e^2=h,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,e^b=eh,g^b=ghi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402544);