#411162

order	512
nilpotency class	3
isoclinism	isoclinic class of #394746
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318358);