#395585

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂×#3018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2138 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)(5,6,11,14,12,16,10,15)(7,17,21,32,22,31,20,13)(8,23,26,29,27,28,25,9)(18,30,24,36,34,35,33,19);(1,2,3,4)(5,7,10,20,12,22,11,21)(6,13,14,17,16,32,15,31)(8,24,25,18,27,33,26,34)(9,19,23,30,29,36,28,35);(5,8)(6,9)(7,18)(10,26)(11,25)(12,27)(13,30)(14,28)(15,23)(16,29)(17,19)(20,24)(21,33)(22,34)(31,36)(32,35);(5,9)(6,8)(7,19)(10,23)(11,28)(12,29)(13,24)(14,25)(15,26)(16,27)(17,18)(20,30)(21,35)(22,36)(31,34)(32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302781);