#385391

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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×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

934 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)(5,6,12,17,10,16,11,18)(7,14,8,22,20,13,21,30)(9,24,28,15,26,33,27,32)(19,34,23,31,25,36,35,29);(1,2)(3,4)(5,7)(6,13)(8,12)(9,25)(10,20)(11,21)(14,16)(15,31)(17,30)(18,22)(19,26)(23,27)(24,34)(28,35)(29,32)(33,36);(5,8,11,7,10,21,12,20)(6,14,17,22,16,13,18,30)(9,23,27,19,26,35,28,25)(15,31,33,36,32,29,24,34);(1,3,4,2)(5,9,12,28,10,26,11,27)(6,15,18,24,16,32,17,33)(7,19,8,23,20,25,21,35)(13,29,22,36,14,31,30,34)

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=ef,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,g^c=gi

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, 292587);