#394595

order	512
nilpotency class	3
isoclinism	isoclinic class of #394590
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3010	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2124 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:

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

Polycyclic presentation:

2^9|c^2=e,d^2=i,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,g^c=gi,h^c=hi,f^d=fi

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, 301791);