#380557

order	512
nilpotency class	3
isoclinism	isoclinic class of #214664
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1513 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,h^b=hi,d^c=de,f^d=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, 287753);