#464748

order	512
nilpotency class	3
isoclinism	isoclinic class of #454982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1540 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)(7,11,17,25)(8,12)(9,14)(10,13)(15,21)(16,27,31,23)(18,24,33,26)(19,29)(20,30)(22,28)(32,35,34,36);(1,2,3,4)(5,7,9,17)(6,11,14,25)(8,16,19,31)(10,18,21,33)(12,27,29,23)(13,24,15,26)(20,32,22,34)(28,36,30,35);(2,4)(5,8,9,19)(6,12,14,29)(7,16,17,31)(10,22,21,20)(11,23,25,27)(13,28,15,30)(18,34,33,32)(24,35,26,36);(6,13,14,15)(8,19)(11,24,25,26)(12,28,29,30)(16,31)(20,22)(23,35,27,36)(32,34)

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=i,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,h^c=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, 371944);