#394592

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	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2164 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:

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

Polycyclic presentation:

2^9|a^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, 301788);