#378905

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

975 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286101);