#356683

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	#2361
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1517 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)(5,6,10,17,12,18,11,16)(7,19,22,13,24,30,23,31)(8,25,26,14,27,9,15,29)(20,34,35,33,36,21,32,28);(1,2)(3,4)(5,7,11,23,12,24,10,22)(6,13,17,30,18,31,16,19)(8,20,15,32,27,36,26,35)(9,28,29,34,25,33,14,21);(5,8,10,26,12,27,11,15)(6,14,17,9,18,29,16,25)(7,20,22,35,24,36,23,32)(13,21,30,28,31,34,19,33);(2,4)(5,9,10,29,12,25,11,14)(6,15,17,8,18,26,16,27)(7,21,22,28,24,34,23,33)(13,20,30,35,31,36,19,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=dei,f^c=fi,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, 263879);