#395398

order	512
nilpotency class	3
isoclinism	isoclinic class of #393921
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	#62842	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3640 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:

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

Polycyclic presentation:

2^9|a^2=ef,e^2=i,h^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,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, 302594);