#358720

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,10)(8,9)(11,12)(13,14,18,24,19,22,17,23)(15,25,28,36,29,20,27,37)(16,30,33,39,34,21,32,40)(26,35,42,44,43,41,31,38);(1,2)(3,4)(6,9)(10,12)(13,15,19,29)(14,20,22,25)(16,31,34,42)(17,28,18,27)(21,38,30,44)(23,37,24,36)(26,33,43,32)(35,40,41,39);(2,4)(13,16,19,34)(14,21,22,30)(15,26,29,43)(17,33,18,32)(20,35,25,41)(23,40,24,39)(27,42,28,31)(36,44,37,38);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265916);