#461425

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1502 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368621);