#383556

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	D₄×C₂²⋊C₄
prime radical	#60453	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1409 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=e,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^b=ei,f^c=fi,f^d=fi

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, 290752);