#412015

order	512
nilpotency class	3
isoclinism	isoclinic class of #395246
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1368 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)(8,20)(9,10)(12,25)(13,28)(15,23)(17,31)(18,24)(21,22)(26,30)(29,33)(32,34)(35,36);(1,2)(3,4)(5,6)(7,16)(8,13)(9,23)(10,15)(11,14)(12,18)(17,30)(19,27)(20,28)(21,33)(22,29)(24,25)(26,31)(32,36)(34,35);(1,3,4,2)(5,7)(6,12)(8,17)(9,24)(10,18)(11,19)(13,26)(14,25)(15,27)(16,23)(20,31)(21,34)(22,32)(28,36)(29,35)(30,33);(5,8,10,20,11,22,9,21)(6,13,15,28,14,29,23,33)(7,17,18,31,19,32,24,34)(12,26,27,36,25,35,16,30)

Polycyclic presentation:

2^9|c^2=e,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,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, 319211);