#148891

order	512
nilpotency class	4
isoclinism	isoclinic class of #134691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1808
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

464 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=hi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,g^c=gi

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, 56087);