#504675

order	512
nilpotency class	4
isoclinism	isoclinic class of #504384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1753 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

523 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=ei,d^2=fhi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=gh,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411871);