#457756

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

841 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)(5,6,10,15,9,12,11,16)(7,17,18,26,21,29,20,13)(8,22,24,14,23,30,25,27)(19,33,34,31,35,28,32,36);(6,12)(7,18,21,20)(13,17,26,29)(14,27)(15,16)(19,32,35,34)(22,30)(28,31,33,36);(5,7,11,18,9,21,10,20)(6,13,16,17,12,26,15,29)(8,19,25,34,23,35,24,32)(14,28,22,36,27,33,30,31);(2,4)(5,8)(6,14)(7,19)(9,23)(10,25)(11,24)(12,27)(13,28)(15,22)(16,30)(17,31)(18,32)(20,34)(21,35)(26,33)(29,36)

Polycyclic presentation:

2^9|a^2=f,b^2=ei,c^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 364952);