#353526

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1774 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,d^b=di,g^b=gi,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210945, #72210978, #72210987, …
maximal normal subgroup of	#72210945, #72210978, #72210987, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260722);