#464078

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1465 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)(5,6,11,16)(7,17,21,27)(8,22,24,13)(9,15,10,14)(12,20,28,18)(19,31,34,36)(23,29,25,30)(26,33,35,32);(5,7,10,20,11,21,9,18)(6,12,15,27,16,28,14,17)(8,19,23,33,24,34,25,32)(13,26,29,36,22,35,30,31);(2,4)(7,18)(8,23)(9,10)(12,17)(13,29)(14,15)(20,21)(22,30)(24,25)(27,28)(31,36)(32,33);(5,8)(6,13)(7,19)(9,25)(10,23)(11,24)(12,26)(14,30)(15,29)(16,22)(17,31)(18,32)(20,33)(21,34)(27,36)(28,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371274);