#494388

order	512
nilpotency class	4
isoclinism	isoclinic class of #33568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 401584);