#355479

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1508 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)(7,15,18,11)(8,19,21,30)(9,14)(10,13)(12,22,28,23)(16,33)(17,31,24,27)(20,36)(25,29)(26,34)(32,35);(1,2,3,4)(5,7)(6,11)(8,20)(9,17)(10,18)(12,26)(13,15)(14,27)(16,21)(19,36)(22,35)(23,34)(24,25)(28,32)(29,31)(30,33);(5,8,9,23,10,21,25,22)(6,12,14,30,13,28,29,19)(7,16,24,34,18,20,17,35)(11,26,31,36,15,32,27,33);(5,7)(6,11)(8,16)(9,24)(10,18)(12,26)(13,15)(14,31)(17,25)(19,33)(20,21)(22,35)(23,34)(27,29)(28,32)(30,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,f^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, 262675);