#121017

order	512
nilpotency class	3
isoclinism	isoclinic class of #10386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,7,10)(3,11)(4,5,9,6)(8,13)(12,16)(14,15)(17,18,22,26,23,28,21,27)(19,29,31,24,33,40,32,41)(20,34,36,44,37,43,38,25)(30,42,47,48,35,39,46,45);(1,3,6,13,7,14,5,12)(2,8,9,11,10,16,4,15)(17,19,21,32,23,33,22,31)(18,24,27,29,28,41,26,40)(20,35,36,46,37,30,38,47)(25,42,34,48,44,39,43,45);(1,4,5,2,7,9,6,10)(3,11,12,16,14,15,13,8)(17,20,21,36,23,37,22,38)(18,25,26,43,28,44,27,34)(19,30,31,46,33,35,32,47)(24,39,29,45,41,42,40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28213);