#471437

order	512
nilpotency class	3
isoclinism	isoclinic class of #455822
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

1077 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)(5,6,10,8)(7,15,18,12)(9,22,23,28)(11,21,16,14)(13,25,20,24)(17,32,33,36)(19,27,31,30)(26,34,29,35);(5,7,16,31,10,18,11,19)(6,12,21,30,8,15,14,27)(9,17,24,34,23,33,25,35)(13,26,22,36,20,29,28,32);(2,4)(5,8,10,6)(7,15,18,12)(9,20,23,13)(11,14,16,21)(17,26,33,29)(19,27,31,30)(22,25,28,24)(32,35,36,34);(5,9)(6,13)(7,17)(8,20)(10,23)(11,25)(12,26)(14,28)(15,29)(16,24)(18,33)(19,35)(21,22)(27,32)(30,36)(31,34)

Polycyclic presentation:

2^9|a^2=i,b^2=egh,c^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,h^c=hi,f^d=fi,g^d=gi,h^d=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, 378633);