#465831

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1043 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,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, 373027);