#495760

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	C₂×#454	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1057 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)(7,16)(8,20)(9,12)(10,15)(11,18)(13,23)(14,25)(17,31)(19,33)(21,28)(22,30)(24,34)(26,32)(27,35)(29,36);(5,7,10,17)(6,11,15,28)(8,21,23,18)(9,24,25,19)(12,27,14,29)(13,31,20,16)(22,35,26,36)(30,33,32,34);(6,12)(9,25)(11,27)(13,32)(14,15)(16,33)(19,24)(20,30)(22,26)(28,29)(31,34)(35,36);(1,3,4,2)(7,17)(11,28)(16,31)(18,21)(19,24)(27,29)(33,34)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=g,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, 402956);