#465834

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

1042 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 373030);