#470822

order	512
nilpotency class	3
isoclinism	isoclinic class of #460861
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

700 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:

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

Polycyclic presentation:

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

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, 378018);