#449076

order	512
nilpotency class	3
isoclinism	isoclinic class of #421163
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

847 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)(7,12,17,27)(8,21)(9,13)(10,16)(11,15)(14,24)(18,33,32,36)(19,28,20,25)(22,29)(23,31)(26,34,30,35);(5,7,9,19,11,17,10,20)(6,12,16,25,15,27,13,28)(8,18,23,34,22,32,24,35)(14,30,29,33,31,26,21,36);(6,13)(7,17)(9,10)(12,25)(14,29)(15,16)(18,32)(21,31)(23,24)(26,33)(27,28)(30,36);(2,4)(5,8)(6,14)(7,18,17,32)(9,24)(10,23)(11,22)(12,26,27,30)(13,29)(15,31)(16,21)(19,35,20,34)(25,33,28,36)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=g,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,d^b=de,e^b=ei,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356272);