#494306

order	512
nilpotency class	4
isoclinism	isoclinic class of #2190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#447
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,e^b=eh,f^b=fhi,h^b=hi,e^c=ehi,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401502);