#396837

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

953 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)(5,6,12,16)(7,17,21,27)(8,22,24,30)(9,10,15,11)(13,20,28,19)(14,25,29,23)(18,32,34,26)(31,35,36,33);(5,7,11,20,12,21,10,19)(6,13,15,27,16,28,9,17)(8,18,23,33,24,34,25,35)(14,26,22,31,29,32,30,36);(5,8)(6,14)(7,18)(9,22)(10,23)(11,25)(12,24)(13,26)(15,30)(16,29)(17,31)(19,33)(20,35)(21,34)(27,36)(28,32);(2,4)(5,9,10,16,12,15,11,6)(7,17,19,28,21,27,20,13)(8,22,23,29,24,30,25,14)(18,31,33,32,34,36,35,26)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304033);