#119660

order	512
nilpotency class	3
isoclinism	isoclinic class of #10637
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

686 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=hi,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,f^b=fh,g^b=gh,e^c=ei,f^c=fh,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26856);