#457758

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

852 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)(5,6,9,14,11,15,10,16)(7,12,18,27,19,29,20,28)(8,21,24,30,23,31,25,13)(17,32,33,36,22,35,34,26);(5,7)(6,12)(8,22,23,17)(9,18)(10,20)(11,19)(13,26,30,36)(14,27)(15,29)(16,28)(21,32,31,35)(24,34,25,33);(5,8,9,25,11,23,10,24)(6,13,14,31,15,30,16,21)(7,17,18,34,19,22,20,33)(12,26,27,35,29,36,28,32);(2,4)(6,14)(8,23)(9,10)(12,27)(13,21)(15,16)(17,22)(18,20)(26,32)(28,29)(30,31)(35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,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, 364954);