#419910

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

1538 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)(5,6,10,16,12,17,11,15)(7,18,22,31,23,32,21,13)(8,24,27,30,28,9,26,14)(19,29,25,20,34,35,36,33);(5,7)(6,13)(8,25)(9,29)(10,22)(11,21)(12,23)(14,20)(15,32)(16,18)(17,31)(19,26)(24,35)(27,34)(28,36)(30,33);(5,8,12,28)(6,14,17,30)(7,19,23,34)(9,15,24,16)(10,26,11,27)(13,29,31,35)(18,33,32,20)(21,25,22,36);(2,4)(5,9,10,14,12,24,11,30)(6,8,15,26,17,28,16,27)(7,20,21,29,23,33,22,35)(13,19,18,25,31,34,32,36)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 327106);