#434828

order	512
nilpotency class	3
isoclinism	isoclinic class of #419839
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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

998 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)(5,6,9,15,11,17,10,16)(7,12,24,14,20,28,19,27)(8,21,23,30,22,31,25,13)(18,29,34,26,32,35,33,36);(5,7)(6,12)(8,18,22,32)(9,24)(10,19)(11,20)(13,26,30,36)(14,15)(16,27)(17,28)(21,35,31,29)(23,34,25,33);(5,8,9,25,11,22,10,23)(6,13,15,31,17,30,16,21)(7,18,19,34,20,32,24,33)(12,26,27,35,28,36,14,29);(2,4)(5,7)(6,14)(8,18)(9,19)(10,24)(11,20)(12,15)(13,29)(16,28)(17,27)(21,36)(22,32)(23,33)(25,34)(26,31)(30,35)

Polycyclic presentation:

2^9|a^2=efg,b^2=e,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,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, 342024);