#465626

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1134 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)(7,12)(8,21)(9,16)(10,15)(11,17)(13,19)(14,25)(18,31)(20,29)(22,30)(23,33)(24,32)(26,28)(27,34)(35,36);(1,2,3,4)(5,7)(6,12)(8,18)(9,19)(10,26)(11,20)(13,16)(14,27)(15,28)(17,29)(21,30)(22,23)(24,34)(25,36)(31,33)(32,35);(2,4)(5,7)(6,13)(8,22)(9,26)(10,19)(11,20)(12,16)(14,31)(15,29)(17,28)(18,23)(21,35)(24,34)(25,36)(27,33)(30,32);(5,8,9,24,11,23,10,25)(6,14,15,33,17,32,16,21)(7,18,19,34,20,22,26,36)(12,27,28,31,29,35,13,30)

Polycyclic presentation:

2^9|b^2=g,d^2=f,f^2=i,h^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

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, 372822);