#459318

order	512
nilpotency class	3
isoclinism	isoclinic class of #453999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1165 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:

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

Polycyclic presentation:

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

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, 366514);