#380768

order	512
nilpotency class	3
isoclinism	isoclinic class of #29751
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287964);