#420412

order	512
nilpotency class	3
isoclinism	isoclinic class of #419792
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1646 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:

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

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,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi

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, 327608);