#420049

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1328 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 327245);