#442253

order	512
nilpotency class	3
isoclinism	isoclinic class of #421903
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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eg,c^2=ei,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^c=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, 349449);