#410691

order	512
nilpotency class	3
isoclinism	isoclinic class of #393988
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 317887);