#463479

order	512
nilpotency class	3
isoclinism	isoclinic class of #455641
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

747 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)(5,6,12,15,11,17,10,16)(7,18,21,13,23,31,22,32)(8,9,25,28,27,29,26,14)(19,30,36,20,24,35,34,33);(5,7)(6,13)(8,24)(9,20)(10,22)(11,23)(12,21)(14,30)(15,31)(16,18)(17,32)(19,27)(25,34)(26,36)(28,35)(29,33);(5,8,10,26,11,27,12,25)(6,14,15,9,17,28,16,29)(7,19,22,34,23,24,21,36)(13,30,31,20,32,35,18,33);(2,4)(5,9,10,28,11,29,12,14)(6,8,15,26,17,27,16,25)(7,20,22,35,23,33,21,30)(13,24,31,36,32,19,18,34)

Polycyclic presentation:

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

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, 370675);