#465714

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

898 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,h^b=hi,d^c=de,f^c=fi,e^d=ei,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, 372910);