#119760

order	512
nilpotency class	3
isoclinism	isoclinic class of #10508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

594 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:

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

Polycyclic presentation:

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

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, 26956);