#368657

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2366
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 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)(5,6,12,15,11,16,10,28)(7,17,22,31,21,32,20,13)(8,23,26,35,25,14,9,27)(18,30,34,29,19,33,24,36);(5,7,11,21)(6,13)(8,24,25,34)(9,19,26,18)(10,20,12,22)(14,30)(15,17)(16,31)(23,33)(27,29)(28,32)(35,36);(2,4)(5,8,12,26,11,25,10,9)(6,14,15,27,16,23,28,35)(7,18,20,24,21,19,22,34)(13,29,32,30,31,36,17,33);(5,9,10,25,11,26,12,8)(6,14,15,27,16,23,28,35)(7,19,20,34,21,18,22,24)(13,30,17,29,31,33,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275853);