#355484

order	512
nilpotency class	3
isoclinism	isoclinic class of #21782
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1510 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fgi,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=ce,g^b=gi,e^c=ei,f^c=fi,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262680);