#355482

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1448 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,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, 262678);