#149259

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2087
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,4)(5,6,10,19,13,20,12,17)(7,21,26,52,27,53,24,42)(8,28,31,15,33,47,38,48)(9,34,36,49,37,18,11,16)(14,39,41,60,43,25,50,23)(22,54,57,68,58,61,56,44)(29,62,55,40,59,66,63,67)(30,51,35,46,64,45,32,65);(1,2,3,4)(5,7,11,25,12,24,37,60,13,27,36,39,10,26,9,23)(6,14,34,52,19,41,49,53,20,43,18,42,17,50,16,21)(8,29,35,56,31,55,64,22,33,59,32,57,38,63,30,58)(15,44,46,62,28,61,51,67,48,68,65,66,47,54,45,40);(5,8,10,31,13,33,12,38)(6,15,17,28,20,48,19,47)(7,22,24,56,27,58,26,57)(9,35,36,64,37,32,11,30)(14,40,41,66,43,67,50,62)(16,46,18,51,49,65,34,45)(21,44,52,54,53,68,42,61)(23,59,25,55,60,29,39,63)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56455);