#465627

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1086 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,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, 372823);