#472281

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

991 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,g^d=gi,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, 379477);