#108423

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₄×C₂²⋊C₄
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=ci,e^c=eh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15619);