#412337

order	512
nilpotency class	3
isoclinism	isoclinic class of #396373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1234 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319533);