#461429

order	512
nilpotency class	3
isoclinism	isoclinic class of #454664
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)(5,6,10,15,11,12,9,24)(7,16,17,26,19,28,20,13)(8,14,22,31,23,27,25,30)(18,32,33,36,21,35,34,29);(6,12)(7,17)(9,10)(13,28)(14,27)(16,26)(18,33)(19,20)(21,34)(22,25)(29,35)(32,36);(5,7,9,20,11,19,10,17)(6,13,15,16,12,26,24,28)(8,21,22,34,23,18,25,33)(14,29,30,35,27,36,31,32);(2,4)(5,8)(6,14)(7,18)(9,25)(10,22)(11,23)(12,27)(13,29)(15,31)(16,32)(17,33)(19,21)(20,34)(24,30)(26,36)(28,35)

Polycyclic presentation:

2^9|a^2=g,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 368625);