#494391

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	C₂²×#41	D₄
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1109 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)(7,16)(8,11)(9,22)(10,15)(12,19)(13,20)(14,23)(17,31)(18,32)(21,30)(24,28)(25,27)(26,35)(29,36)(33,34);(1,3,4,2)(6,11)(8,20)(9,23)(12,26)(13,15)(14,27)(16,32)(18,34)(19,24)(22,30)(28,29)(31,33);(5,7)(6,12)(8,18)(9,24)(10,17)(11,26)(13,29)(14,31)(15,28)(16,22)(19,23)(20,34)(21,36)(25,35)(27,33)(30,32);(7,17)(12,28)(16,31)(18,34)(19,24)(26,29)(32,33)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,e^b=ei,f^b=fhi,d^c=di,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, 401587);