#116271

order	512
nilpotency class	3
isoclinism	isoclinic class of #10378
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#393
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

706 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,e^2=hi,f^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,f^b=fhi,d^c=dh,e^c=ehi,f^c=fi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23467);