#394591

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

2126 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^c=fi,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, 301787);