#464232

order	512
nilpotency class	3
isoclinism	isoclinic class of #455155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1798 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:

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

Polycyclic presentation:

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

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, 371428);