#476523

order	512
nilpotency class	3
isoclinism	isoclinic class of #476175
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1337 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383719);