#383512

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	#60422	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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,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, 290708);