#355104

order	512
nilpotency class	3
isoclinism	isoclinic class of #21782
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1933 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)(7,14,9,12)(8,19,22,32)(10,16)(11,15)(13,21,29,24)(17,33)(18,30,23,27)(20,36)(25,31)(26,34)(28,35);(1,2,3,4)(5,7,11,9)(6,12,15,14)(8,17,22,20)(10,23,25,18)(13,28,29,26)(16,30,31,27)(19,36,32,33)(21,35,24,34);(5,8)(6,13)(7,17)(9,20)(10,24)(11,22)(12,26)(14,28)(15,29)(16,32)(18,35)(19,31)(21,25)(23,34)(27,33)(30,36);(5,9,11,7)(6,14,15,12)(8,20,22,17)(10,23,25,18)(13,28,29,26)(16,30,31,27)(19,33,32,36)(21,35,24,34)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=ce,f^b=fi,g^b=gi,f^c=fi,g^c=gi,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, 262300);