#146788

order	512
nilpotency class	4
isoclinism	isoclinic class of #2190
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1786
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dhi,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250656, #72250659, #72250672, …
maximal normal subgroup of	#72250656, #72250659, #72250672, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53984);