#382358

order	512
nilpotency class	3
isoclinism	isoclinic class of #30236
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

683 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289554);