#435251

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2571
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

659 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=ef,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,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, 342447);