#422932

order	512
nilpotency class	3
isoclinism	isoclinic class of #418504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1931 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252190, #72256181, #72256186
maximal normal subgroup of	#72252190, #72256181, #72256186

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330128);