#384420

order	512
nilpotency class	3
isoclinism	isoclinic class of #232128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

960 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ef,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^d=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, 291616);