#395470

order	512
nilpotency class	3
isoclinism	isoclinic class of #394046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62859	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3444 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302666);