#410668

order	512
nilpotency class	3
isoclinism	isoclinic class of #393899
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×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1078 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)(7,19,23,36)(8,20,26,32)(9,15)(10,16)(11,17)(12,18)(13,21,31,22)(14,25,33,30)(24,27)(28,34)(29,35);(1,2,4,3)(5,7)(6,13)(8,24)(9,20)(10,22)(11,21)(12,23)(14,27)(15,25)(16,36)(17,19)(18,31)(26,34)(28,33)(29,32)(30,35);(1,2,4,3)(5,8,12,26)(6,14)(7,15)(9,13,29,31)(10,30,11,25)(16,32)(17,20)(18,33)(19,27,36,28)(21,34)(22,24)(23,35);(1,3,4,2)(5,9,11,28,12,29,10,27)(6,15,17,34,18,35,16,24)(7,20,21,33,23,32,22,14)(8,13,25,19,26,31,30,36)

Polycyclic presentation:

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

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, 317864);