#471436

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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1133 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)(5,6,9,13)(7,15,18,11)(8,20,21,28)(10,23,16,14)(12,24,27,22)(17,32,33,36)(19,26,31,30)(25,34,29,35);(5,7,16,31,9,18,10,19)(6,11,23,30,13,15,14,26)(8,17,22,34,21,33,24,35)(12,25,20,36,27,29,28,32);(2,4)(10,16)(14,23)(17,33)(19,31)(20,28)(22,24)(25,29)(26,30);(5,8)(6,12)(7,17)(9,21)(10,24)(11,25)(13,27)(14,28)(15,29)(16,22)(18,33)(19,35)(20,23)(26,32)(30,36)(31,34)

Polycyclic presentation:

2^9|a^2=i,b^2=egh,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, 378632);