#356727

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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1212 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:

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

Polycyclic presentation:

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

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, 263923);