#358443

order	512
nilpotency class	3
isoclinism	isoclinic class of #23414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#392
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1930 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,c^b=cg,f^b=fi,e^c=ei

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, 265639);