#385244

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	D₄×C₄⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

936 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292440);