#420403

order	512
nilpotency class	3
isoclinism	isoclinic class of #418508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1642 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=e,d^2=e,e^2=i,f^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,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, 327599);