#411160

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=g,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=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, 318356);