#442132

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
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

937 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)(5,6,12,16,11,17,10,18)(7,19,8,13,22,29,21,14)(9,24,27,15,26,32,25,33)(20,28,23,36,35,30,34,31);(5,7,12,8,11,22,10,21)(6,13,16,29,17,14,18,19)(9,23,27,35,26,34,25,20)(15,31,32,28,33,36,24,30);(5,8,10,22,11,21,12,7)(6,14,18,19,17,13,16,29)(9,23,25,35,26,34,27,20)(15,30,24,31,33,28,32,36);(2,4)(5,9,12,27,11,26,10,25)(6,15,18,24,17,33,16,32)(7,20,21,34,22,35,8,23)(13,28,29,36,14,30,19,31)

Polycyclic presentation:

2^9|a^2=efg,b^2=efg,c^2=ef,d^2=ei,e^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,h^b=hi,g^c=gi

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, 349328);