#146387

order	512
nilpotency class	3
isoclinism	isoclinic class of #1795
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53583);