#461305

order	512
nilpotency class	3
isoclinism	isoclinic class of #454986
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

865 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=i,d^2=e,g^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,d^c=de,g^c=gi,h^c=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, 368501);