#392596

order	512
nilpotency class	3
isoclinism	isoclinic class of #32894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3385 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299792);