#356996

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2366
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1426 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=fg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=dei,f^c=fi,g^c=gi,e^d=ei,h^d=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264192);