#182216

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1008 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,e^2=h,g^2=i,b^a=be,d^a=di,e^a=eh,g^a=gh,c^b=ci,d^c=dh

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, 89412);