#460611

order	512
nilpotency class	3
isoclinism	isoclinic class of #454676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

972 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367807);