#453402

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

617 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=f,c^2=eg,d^2=e,e^2=i,g^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,f^c=fi

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, 360598);