#148941

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1942
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

434 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,f^b=fh,h^b=hi,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, 56137);