#439070

order	512
nilpotency class	3
isoclinism	isoclinic class of #422233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1379 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 346266);