#119595

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=h,e^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,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, 26791);