#461831

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=efi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,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, 369027);