#420484

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1632 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)(5,6,10,28,11,16,23,15)(7,17,29,31,21,34,20,12)(8,13,32,9,24,26,14,27)(18,25,33,19,22,35,36,30);(2,3)(5,7)(6,12)(8,22)(9,25)(10,29)(11,21)(13,30)(14,33)(15,34)(16,31)(17,28)(18,24)(19,26)(20,23)(27,35)(32,36);(1,2,4,3)(5,8,23,32,11,24,10,14)(6,13,28,27,16,26,15,9)(7,18,29,36,21,22,20,33)(12,30,34,25,31,19,17,35);(1,3,4,2)(5,9,10,26,11,27,23,13)(6,14,15,24,16,32,28,8)(7,19,20,25,21,30,29,35)(12,18,17,33,31,22,34,36)

Polycyclic presentation:

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

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, 327680);