#356356

order	512
nilpotency class	3
isoclinism	isoclinic class of #3288
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×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	D₄×C₄⋊C₄
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

974 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)(7,17,22,33)(8,23)(9,13)(10,14)(11,16)(12,20,30,21)(15,27)(18,35,19,28)(24,31)(25,34,36,29)(26,32);(1,2,3,4)(5,7)(6,12)(8,18)(9,25)(10,21)(11,22)(13,28)(14,33)(15,17)(16,30)(19,24)(20,27)(23,34)(26,36)(29,31)(32,35);(5,8,10,9,11,24,27,26)(6,13,15,23,16,32,14,31)(7,18,21,25,22,19,20,36)(12,28,17,34,30,35,33,29);(5,9,10,24,11,26,27,8)(6,13,14,31,16,32,15,23)(7,19,20,25,22,18,21,36)(12,29,17,28,30,34,33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263552);