#471041

order	512
nilpotency class	3
isoclinism	isoclinic class of #459444
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

818 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:

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

Polycyclic presentation:

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

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, 378237);